MATHEMATICAL MODELS IN EPIDEMIOLOGY. 9781493998265, 1493998269

Table of contents :
Introduction
A prelude to mathematical epidemiology.- Simple compartmental models for disease transmission.- Endemic disease models.- Epidemic models.- Models with heterogeneous mixing.- Models for diseases transmitted by vectors.- models for tuberculosis.- Models for HIV?AIDS.- Models for influenza.- Models for Ebola.- Models for malaria.- Dengue fever and the Zika virus.- Disease transmission models withagedpendence.- Spatial structure in disease transmission models.- Epidemiological models incorporating mobility, behavior, and time scales.- Challenges, opportunities, and theoretical epidemiology.

Citation preview

Texts in Applied Mathematics 69

Fred Brauer Carlos Castillo-Chavez Zhilan Feng

Mathematical Models in Epidemiology

Texts in Applied Mathematics Volume 69

Series Editors Anthony Bloch, University of Michigan, Ann Arbor, MI, USA Charles L. Epstein, University of Pennsylvania, Philadelphia, PA, USA Alain Goriely, University of Oxford, Oxford, UK Leslie Greengard, New York University, New York, NY, USA Advisory Board J. Bell, Lawrence Berkeley National Lab, Berkeley, USA R. Kohn, New York University, New York, USA P. Newton, University of Southern California, Los Angeles, USA C. Peskin, New York University, New York, USA R. Pego, Carnegie Mellon University, Pittsburgh, USA L. Ryzhik, Stanford University, Stanford, USA A. Singer, Princeton University, Princeton, USA A. Stevens, Max-Planck-Institute for Mathematics, Leipzig, Germany A. Stuart, University of Warwick, Coventry, UK T. Witelski, Duke University, Durham, USA S. Wright, University of Wisconsin, Madison, USA

The mathematization of all sciences, the fading of traditional scientific boundaries, the impact of computer technology, the growing importance of computer modeling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The aim of this series is to provide such textbooks in applied mathematics for the student scientist. Books should be well illustrated and have clear exposition and sound pedagogy. Large number of examples and exercises at varying levels are recommended. TAM publishes textbooks suitable for advanced undergraduate and beginning graduate courses, and complements the Applied Mathematical Sciences (AMS) series, which focuses on advanced textbooks and research-level monographs.

More information about this series at http://www.springer.com/series/1214

Fred Brauer • Carlos Castillo-Chavez • Zhilan Feng

Mathematical Models in Epidemiology

123

Fred Brauer University of British Columbia Department of Mathematics Vancouver, BC, Canada Zhilan Feng Purdue University Department of Mathematics West Lafayette, IN, USA

Carlos Castillo-Chavez Mathematical and Computational Modeling Center (MCMSC) Department of Mathematics and Statistics Arizona State University Tempe, AZ, USA

ISSN 0939-2475 ISSN 2196-9949 (electronic) Texts in Applied Mathematics ISBN 978-1-4939-9826-5 ISBN 978-1-4939-9828-9 (eBook) https://doi.org/10.1007/978-1-4939-9828-9 Mathematics Subject Classification: 92D30 © Springer Science+Business Media, LLC, part of Springer Nature 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Science+Business Media, LLC, part of Springer Nature. The registered company address is: 233 Spring Street, New York, NY 10013, U.S.A.

Foreword

The subject of infectious diseases has been one of the richest areas of application of mathematics in biology, dating back to Sir Ronald Ross’ classic contributions to the management of malaria at the beginning of the last century. Infectious diseases obviously impose great burdens of morbidity and mortality on humanity, as well as on non-human populations of importance to us; and hence, the ability to use formal models to reduce those burdens has attracted the attention of exceptional mathematicians, including all three of the authors of this volume. All have made major contributions to the subject, independently and in collaboration with each other, and the current volume profits from their complementary talents and expertise. The authors combine broad expertise built on excellent scholarship, their own fundamental research and masterful exposition, and I am delighted to have been asked to write this Foreword. The subject of mathematical epidemiology has seen great progress in the century since Ross, as new diseases like SARS and HIV–AIDS have appeared and as older ones, like measles and TB, have waned and waxed and re-emerged in places where they were thought to be defeated. But our battle with infectious diseases can never be won completely; like Lewis Carroll’s Red Queen said, we must keep running just “to keep in the same place”; and we will need to run even faster if we are to make progress in this perpetual struggle. Highly adaptive agents like influenza A continue to evolve in a continual struggle with immune defenses, fostering increased efforts to develop a universal vaccine; similarly, pathogenic bacteria have evolved resistance to the drugs that had at first tamed them. At the same time, a growing and increasingly mobile world population has accelerated the spread of disease, including novel diseases that previously would have remained endemic or died out. Climate change has added yet another challenge, as the vectors of diseases, from insects to migrant birds, shift their ranges to carry diseases into areas whose populations are unprepared to deal with them. As discussed in the next paragraph, social influences raise new challenges for management and for mathematical modeling. Even our own microbiomes have been dramatically affected by environmental influences, antibiotic uses, and medical practices such as C-sections, changing the background against which management must operate. v

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Foreword

Medical science has responded brilliantly to these challenges, through the development of new antibiotics, vaccines, and treatment protocols, guided along the way by improved mathematical modeling. Such models have needed to address the multi-scale dynamics of an interacting system of within-host immune dynamics, the population-level modeling of disease spread, the dynamics of vector populations, and the socioeconomic context within which diseases spread. The socioeconomic issues include not only the economics of control measures but also the behavioral responses before and during outbreaks, from vaccine hesitancy to the avoidance of social contacts involving infectious individuals. The overuse of antibiotics and underuse of vaccines have created new dilemmas for practitioners and new avenues for mathematical modeling to make contributions, guiding individual decisions, medical practice, hospital decision-making, and public policy at all levels. This volume is a comprehensive treatment of the classical theory and modern extensions of that theory. It is constructed as a textbook for students with solid mathematical foundations, including problem sets that illuminate the issues. As such, it should find wide use as the basis for beginning and advanced courses in this important subject and serve as a complement and update to the excellent 2001 text by Brauer and Castillo-Chavez; clearly, a lot has happened in the field in the nearly two decades since that book appeared, and those advances are well documented in this book. But this volume, as the earlier one, also will be a treasure trove for mathematicians and other researchers looking for an authoritative introduction to the literature and to open problems to the solutions of which they might contribute their skills. It should find its way onto the shelves of every researcher interested in the topic. Princeton University Princeton, NJ, USA

Simon Levin

Preface

The goal of this book is to interest students of mathematics and public health professionals in the modeling of infectious diseases transmission. We believe that some knowledge of disease transmission models can give useful insights to epidemiologists and that there are interesting mathematical problems in such models. The mathematical background necessary for studying this book is a knowledge of calculus, some matrix theory, and some ordinary differential equations, specifically approximate and qualitative methods. Our emphasis is on describing the mathematical results being used and showing how to apply them rather than on detailed proofs. We hope that eventually the education of biologists and public health professionals will include these mathematical topics in the first 2 years of university so that this book can be accessible to upper-level undergraduate students. A course based on this book should cover the first section on basic concepts and some of the chapters on specific diseases in the second section. In addition, some of the more advanced topics from the third section, depending on the interests of the students and the time available, can be included. Throughout the book, there are suggested projects giving a sense of research topics that could be attacked by students in groups. In the first section of the book, there are also numerous exercises, and answers to selected exercises are available. We believe that a book should go beyond a course based on it, to lead to further reading, and the last three chapters of the book and the epilogue are intended to suggest some topics for further thought. In order to make the book more accessible to the readers with minimal mathematical background, we have put some mathematical review material on matrix algebra, differential equations, and systems of differential equations into some appendices to this book. We have also established a website https://mcmsc.asu.edu/content/mathematical-models-epidemiology containing review notes covering some topics in calculus, matrix algebra, and dynamical systems (ordinary differential equations and difference equations). We have also starred difficult exercises and difficult sections which are intended for the readers with strong mathematical backgrounds.

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Preface

Stochastic disease transmission models form an important area that we have largely omitted in order to keep the size of this book within bounds. For readers who may wish to explore this subject, we have included some references near the end of Chap. 1. Discrete disease transmission models are not studied in the main text, but there are several projects introducing the subject in addition to some references near the end of Chap. 1. Our emphasis in this book is on relatively simple models whose goal is to describe general qualitative behavior and establish broad principles. Public health professionals are more concerned with detailed models to make short-term quantitative predictions. Such detailed models are generally difficult or impossible to solve analytically, but the relatively recent development of the availability of highspeed computing has made detailed models available for quantitative predictions and comparison of different possible management strategies. The emphasis in this book is on relatively simple models, but readers should remember that there is another important aspect of the subject. Vancouver, BC, Canada Tempe, AZ, USA West Lafayette, IN, USA

Fred Brauer Carlos Castillo-Chavez Zhilan Feng

Acknowledgments

It is always hard to write acknowledgments because so many have contributed in a multitude of ways to the content of his book, and so we hope that our public recognition in this section is accepted by those who are not mentioned explicitly. Thanks for your understanding. This book, like our textbook Mathematical Models in Population Biology and Epidemiology by FB and CCC, has finally been completed after many detours and delays. This volume has been enriched by the inclusion of a third author, Zhilan Feng (ZF). The three of us have been involved in mathematical epidemiology and its applications for roughly three decades, an effort that was intensified by the establishment of the Mathematical and Theoretical Biology Institute (MTBI) by Carlos Castillo-Chavez (CCC) in 1996. This is a summer institute that has mentored nearly 500 undergraduates (152 have completed a PhD), nearly 180 graduate students, several postdocs, and a dozen young faculty. The energy of MTBI and its regular faculty has been a source of inspiration, energy, and ideas for this project. The staff of the Simon A. Levin Mathematical, Computational, and Modeling Sciences Center (SAL-MCMSC) has been involved in the logistics of the MTBI and of this project. We are particularly thankful to Kamal Barley for his invaluable help in drawing most of the figures; to Baltazar Espinoza and Victor Moreno for their assistance in indexing, in various chapters, and some of the computations and graphics; and to MTBI alumni for their useful comments. In addition, Baltazar wrote much of the material on age distributions in Chap. 13. All books have misprints. This one has many fewer misprints than it might otherwise have had because of the proofreading by Baltazar Espinosa, Ceasar Montalvo, Anarina Murillo, and Marisabel Rodriguez. CCC wants to thank his three children, Carlos W, Gabi, and Melissa, and his wife, Nohora, for their support, ideas, and inspiration. ZF wants to thank her daughter, Haiyun, and son, Henry, for all their love and support. FB wants to thank his wife, Esther; his children, David, Deborah, and Michael; and his grandchildren, Noah, Sophie, Benjamin, and Evan, for their support and tolerance. One of the lessons learned from the MTBI is that a cooperating group can accomplish much more than any of the individuals in the group, and this applies to families even without direct involvement. ix

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Acknowledgments

We would like to restate our appreciation for the deliberate or unsuspecting mentoring that we have received over the years from our undergraduate, graduate, and postdoctoral students, colleagues, collaborators, visitors, and researchers in the field across the world. This book would have not completed without the support that we have received over the years by the National Science Foundation, DOD, and occasionally by the NIH, as well as the Canadian group MITACS. The MITACS (Mathematics of Information Technology and Complex Systems) program in response to the SARS epidemic of 2002–2003 has instigated much cooperation of mathematical modelers and public health professionals to learn to communicate with one another. The support that the SAL-MCMSC has received from the Offices of the President and Provost, the PhD Program in Applied Mathematics in the Life and Social Sciences, and the School of Human Evolution and Social Change at Arizona State University have been critical for this effort. We thank them all.

Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part I Basic Concepts of Mathematical Epidemiology 1

Introduction: A Prelude to Mathematical Epidemiology . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Some History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Beginnings of Compartmental Models . . . . . . . . . . . . . . 1.2.2 Stochastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Developments in Compartmental Models. . . . . . . . . . . . . . . . 1.2.4 Endemic Disease Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Diseases Transmitted by Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Heterogeneity of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Strategic Models and This Volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 5 7 9 11 12 14 15 16

2

Simple Compartmental Models for Disease Transmission . . . . . . . . . . . . . 2.1 Introduction to Compartmental Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The SI S Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The SI R Model with Births and Deaths . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Simple Kermack–McKendrick Epidemic Model . . . . . . . . . . . . . 2.5 Epidemic Models with Deaths due to Disease. . . . . . . . . . . . . . . . . . . . . 2.6 *Project: Discrete Epidemic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 *Project: Pulse Vaccination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 *Project: A Model with Competing Disease Strains . . . . . . . . . . . . . . 2.9 Project: An Epidemic Model in Two Patches. . . . . . . . . . . . . . . . . . . . . . 2.10 Project: Fitting Data for an Influenza Model . . . . . . . . . . . . . . . . . . . . . . 2.11 Project: Social Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 23 26 30 35 43 46 47 49 52 53 54 55 60 xi

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Contents

3

Endemic Disease Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 More Complicated Endemic Disease Models . . . . . . . . . . . . . . . . . . . . . 3.1.1 Exposed Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 A Treatment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Vertical Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Some Applications of the SI R Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Herd Immunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Age at Infection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 The Inter-Epidemic Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 “Epidemic” Approach to Endemic Equilibrium . . . . . . . . . 3.3 Temporary Immunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 *Delay in an SI RS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 A Simple Model with Multiple Endemic Equilibria . . . . . . . . . . . . . . 3.5 A Vaccination Model: Backward Bifurcations . . . . . . . . . . . . . . . . . . . . 3.5.1 The Bifurcation Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 *An SEI R Model with General Disease Stage Distributions . . . . 3.6.1 *Incorporation of Quarantine and Isolation . . . . . . . . . . . . . . 3.6.2 *The Reduced Model of (3.42) Under GDA . . . . . . . . . . . . . 3.6.3 *Comparison of EDM and GDM . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Diseases in Exponentially Growing Populations . . . . . . . . . . . . . . . . . . 3.8 Project: Population Growth and Epidemics . . . . . . . . . . . . . . . . . . . . . . . 3.9 *Project: An Environmentally Driven Infectious Disease . . . . . . . . 3.10 *Project: A Two-Strain Model with Cross Immunity . . . . . . . . . . . . . 3.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 64 64 65 67 68 68 69 71 72 73 75 78 81 87 88 93 95 96 100 102 107 111 112 114

4

Epidemic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 A Branching Process Disease Outbreak Model . . . . . . . . . . . . . . . . . . . 4.1.1 Transmissibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Network and Compartmental Epidemic Models . . . . . . . . . . . . . . . . . . 4.3 More Complicated Epidemic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Exposed Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 A Treatment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 An Influenza Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 A Quarantine–Isolation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 An SIR Model with a General Infectious Period Distribution . . . . 4.5 The Age of Infection Epidemic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 A General SEIR Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 A General Treatment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 A General Quarantine/Isolation Epidemic Model . . . . . . . 4.6 The Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Interpretation of Data and Parametrization . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Models of SI R Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Models of SEIR Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Mean Generation Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 118 123 125 129 129 131 133 134 137 139 141 144 146 148 151 152 154 156

Contents

*Effect of Timing of Control Programs on Epidemic Final Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Directions for Generalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Some Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 *Project: A Discrete Model with Quarantine and Isolation . . . . . . . 4.12 Project: Epidemic Models with Direct and Indirect Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.8

5

6

159 161 161 162 167 171 176

Models with Heterogeneous Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 A Vaccination Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Next Generation Matrix and the Basic Reproduction Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Some More Complicated Examples . . . . . . . . . . . . . . . . . . . . . . 5.3 Heterogeneous Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 *Optimal Vaccine Allocation in Heterogeneous Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 A Heterogeneous Mixing Age of Infection Model . . . . . . . . . . . . . . . . 5.4.1 The Final Size of a Heterogeneous Mixing Epidemic . . . 5.5 Some Warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 *Projects: Reproduction Numbers for Discrete Models . . . . . . . . . . 5.7 *Project: Modeling the Synergy Between HIV and HSV-2. . . . . . . 5.8 Project: Effect of Heterogeneities on Reproduction Numbers . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

202 209 212 216 217 221 223 225

Models for Diseases Transmitted by Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 A Basic Vector Transmission Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 The Basic Reproduction Number. . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 The Initial Exponential Growth Rate . . . . . . . . . . . . . . . . . . . . . 6.3 Fast and Slow Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Singular Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 A Vector-Borne Epidemic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 A Final Size Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 *Project: An SEI R/SEI Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 *Project: Models for Onchocerciasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229 229 230 232 233 234 237 239 240 241 241 244 244

179 179 182 188 188

Part II Models for Specific Diseases 7

Models for Tuberculosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 A One-Strain Model with Treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 A Two-Strain TB Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Optimal Treatment Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 251 252 256

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7.4 Modeling of the Long and Variable Latency of TB . . . . . . . . . . . . . . . 7.5 Backward Bifurcation in a TB Model with Reinfection . . . . . . . . . . 7.6 Other TB Models with More Complexities . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Project: Some Calculations for the Two-Strain Model . . . . . . . . . . . . 7.8 Project: Refinements of the One-Strain Model . . . . . . . . . . . . . . . . . . . . 7.9 Project: Refinements of the Two-Strain Model . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

260 263 265 267 268 269 271

8

Models for HIV/AIDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 A Model with Exponential Waiting Times . . . . . . . . . . . . . . . . . . . . . . . . 8.3 An HIV Model with Arbitrary Incubation Period Distributions . . 8.4 An Age of Infection Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 *HIV and Tuberculosis: Dynamics of Coinfections. . . . . . . . . . . . . . . 8.6 *Modeling the Synergy Between HIV and HSV-2 . . . . . . . . . . . . . . . . 8.6.1 Reproduction Numbers for Individual Diseases . . . . . . . . . 8.6.2 Invasion Reproduction Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Influence of HSV-2 on the Dynamics of HIV . . . . . . . . . . . . 8.7 An HIV Model with Vaccination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 A Model with Antiretroviral Therapy (ART) . . . . . . . . . . . . . . . . . . . . . . 8.9 Project: What If Not All Infectives Progress to AIDS? . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

273 273 275 278 281 284 292 295 297 299 299 301 302 305

9

Models for Influenza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction to Influenza Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 A Basic Influenza Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Vaccination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Antiviral Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Seasonal Influenza Epidemics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Season to Season Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Pandemic Influenza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 A Pandemic Outbreak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Seasonal Outbreaks Following a Pandemic . . . . . . . . . . . . . . 9.6 The Influenza Pandemic of 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 A Tactical Influenza Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Multiple Epidemic Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Parameter Estimation and Forecast of the Fall Wave . . . . 9.7 *An SIQR Model with Multiple Strains with Cross-Immunity. . . 9.7.1 *An SIQR Model with a Single Infectious Class . . . . . . . . 9.7.2 *The Case of Two Strains with Cross-Immunity . . . . . . . . 9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 311 312 316 319 323 326 326 327 328 330 330 331 334 337 338 342 347 348

10

Models for Ebola. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 10.1 Estimation of Initial Growth and Reproduction Numbers . . . . . . . . 351 10.1.1 Early Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

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10.2 10.3

Evaluations of Control Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Legrand Model and Underlying Assumptions . . . . . . . . . . . . . . . . 10.3.1 The Legrand Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 A Simpler System Equivalent to the Legrand Model . . . . 10.4 *Models with Various Assumptions on Stage Transition Times. . 10.5 Slower than Exponential Growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 The Generalized Richards Model . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 The Generalized Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 The IDEA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.4 Models with Decreasing Contact Rates . . . . . . . . . . . . . . . . . . 10.6 Project: Slower than Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Project: Movement Restrictions as a Control Strategy . . . . . . . . . . . . 10.8 Project: Effect of Early Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

360 362 362 364 365 382 383 383 384 384 385 386 387 388

11

Models for Malaria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 A Malaria Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Some Model Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Mosquito Incubation Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Boosting of Immunity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Alternative Forms for the Force of Infection. . . . . . . . . . . . . 11.3 *Coupling Malaria Epidemiology and Sickle-Cell Genetics . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

391 391 395 395 396 397 398 407

12

Dengue Fever and the Zika Virus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Dengue Fever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Calculation of the Basic Reproduction Number . . . . . . . . . 12.2 A Model with Asymptomatic Infectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Calculation of the Basic Reproduction Number . . . . . . . . . 12.3 The Zika Virus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 A Model with Vector and Direct Transmission . . . . . . . . . . . . . . . . . . . . 12.4.1 The Initial Exponential Growth Rate . . . . . . . . . . . . . . . . . . . . . 12.5 A Second Zika Virus Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Project: A Dengue Model with Two Patches . . . . . . . . . . . . . . . . . . . . . . 12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

409 409 411 412 412 414 414 417 419 421 422 422

Part III More Advanced Concepts 13

Disease Transmission Models with Age Structure . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Linear Age Structured Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 The Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Equivalent Formulation as an Integral Equation Model. . . . . . . . . . . 13.5 Equilibria and the Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . 13.6 The Demographic Model with Discrete Age Groups . . . . . . . . . . . . .

429 429 430 432 433 435 436

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13.7 13.8

Nonlinear Age Structured Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Age-Structured Epidemic Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8.1 *Age-Dependent Vaccination in Epidemic Models . . . . . 13.8.2 Pair-Formation in Age-Structured Epidemic Models. . . . 13.9 A Multi-Age-Group Malaria Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.10 *Project: Another Malaria Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.11 Project: A Model Without Vaccination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.12 Project: An SI S Model with Age of Infection Structure . . . . . . . . . 13.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

437 439 440 445 447 448 451 452 453 453

14

Spatial Structure in Disease Transmission Models . . . . . . . . . . . . . . . . . . . . . 14.1 Spatial Structure I: Patch Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Spatial Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2 Patch Models with Travel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.3 Patch Models with Residence Times . . . . . . . . . . . . . . . . . . . . . 14.2 Spatial Structure II: Continuously Distributed Models . . . . . . . . . . . 14.2.1 The Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Nonlinear Reaction–Diffusion Equations . . . . . . . . . . . . . . . . 14.2.3 Disease Spread Models with Diffusion . . . . . . . . . . . . . . . . . . 14.3 Project: A Model with Three Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Project: A Patch Model with Residence Time . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

457 457 457 459 461 464 465 467 468 471 474 475

15

Epidemiological Models Incorporating Mobility, Behavior, and Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 General Lagrangian Epidemic Model in an SIS Setting . . . . . . . . . . 15.3 Assessing the Efficiency of Cordon Sanitaire as a Control Strategy of Ebola. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Formulation of the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 *Mobility and Health Disparities on the Transmission Dynamics of Tuberculosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 *A Two-Patch TB Model with Heterogeneity in Population Through Residence Times in the Patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.2 *Results: The Role of Risk and Mobility on TB Prevalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.3 The Role of Risk as Defined by Direct First Time Transmission Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.4 The Impact of Risk as Defined by Exogenous Reinfection Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 *ZIKA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 *Single Patch Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

477 477 479 481 481 482 485

486 486 487 488 490 491

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15.5.2 15.5.3

*Residence Times and Two-Patch Models . . . . . . . . . . . . . . . 492 What Did We Learn from These Single Outbreak Simulations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 Part IV The Future 16

Challenges, Opportunities and Theoretical Epidemiology . . . . . . . . . . . . . 16.1 Disease and the Global Commons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Contagion and Tipping Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.2 Geographic and Spatial Disease Spread. . . . . . . . . . . . . . . . . . 16.2 Heterogeneity of Mixing, Cross-immunity, and Coinfection . . . . . 16.3 Antibiotic Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 A Lagrangian Approach to Modeling Mobility and Infectious Disease Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Behavior, Economic Epidemiology, and Mobility . . . . . . . . . . . . . . . . 16.5.1 Economic Epidemiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.2 Lagrangian and Economic Epidemiology Models . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

507 508 508 509 510 510 512 513 521 521 523 524

Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 A

Some Properties of Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Vectors and Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 The Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

539 539 540 544 545 546 548

B

First Order Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Radioactive Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Solutions and Direction Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Equations with Variables Separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5 Qualitative Properties of Differential Equations. . . . . . . . . . . . . . . . . . .

553 553 555 557 562 569

C

Systems of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 The Phase Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Linearization of a System at an Equilibrium . . . . . . . . . . . . . . . . . . . . . . C.3 Solution of Linear Systems with Constant Coefficients. . . . . . . . . . . C.4 Stability of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Qualitative Behavior of Solutions of Linear Systems . . . . . . . . . . . . .

579 579 581 584 593 598

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Part I

Basic Concepts of Mathematical Epidemiology

Chapter 1

Introduction: A Prelude to Mathematical Epidemiology

1.1 Introduction Recorded history continuously documents the invasion of populations by infectious agents, some causing many deaths before disappearing, others reappearing in invasions some years later in populations that have acquired some degree of immunity, due to prior exposure to related infectious pathogens. The “Spanish” flu epidemic of 1918–1919 exemplifies the devastating impact of relatively rare pandemics; this one was responsible for about 50,000,000 deaths worldwide, while on the mild side of the spectrum we experience annual influenza seasonal epidemics that cause roughly 35,000 deaths in the USA each year. Communicable diseases have played a significant role in shaping human history. The Black Deaths (probably bubonic plague) spread starting in 1346, first through Asia and moving across Europe repeatedly during the fourteenth century. The Black Death has been estimated to have caused the death of as much as one-third of the population of Europe between 1346 and 1350. The disease reappeared regularly in various regions of Europe for more than 300 years, a notable outbreak being that of the Great Plague of London of 1665–1666. It gradually withdrew from Europe afterwards. Some diseases have become endemic (“permanently” established) in various populations causing a variable number of deaths particularly in countries with inefficient or resource-limited health care systems. Even within the twenty-first century, we see that millions of people die of measles, respiratory infections, diarrhea, and more. Individuals still die in significant numbers from diseases that are no longer considered dangerous, diseases that are easily treated or that have been well managed by resource-rich societies or by nations that invest in public health and prevention systematically. Some highly prevalent old foes include malaria, typhus, cholera, schistosomiasis, and sleeping sickness, endemic diseases in many parts of the world; diseases that have a significant negative impact on the mean life span of a population as well as on the economy of afflicted countries due to their impact © Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_1

3

4

1 Introduction: A Prelude to Mathematical Epidemiology

on the health of the population. The World Health Organization has estimated that in 2011 there were 1,400,000 deaths due to tuberculosis, 1,200,000 deaths due to HIV/AIDS, and 627,000 deaths due to malaria (but other sources have estimated the number of malaria deaths to be more than 1,000,000). In short, HIV, malaria, and TB account for at least 9,000 deaths each day. The impact of vaccines can be dramatic, for example, there were 2,600,000 deaths due to measles in 1980 but only 160,000 by 2011. The development and availability of the measles vaccine led to a reduction in the number of deaths due to this childhood disease of nearly 94%. Epidemiologists, in response to a health emergency or as a result of systematic surveillance, first obtain and analyze observed data. They use data, observations, science, and theory as they work at identifying a pathogen (when unknown) behind an observed disease outbreak or as they proceed to plan or implement policies that ameliorates its impact. Naturally, understanding the causes and modes of transmission of each disease is central to forecasting or mitigating its impact within and across populations at risk. Mathematical models have played a substantial role in both short and long term planning for controlling the dynamics of a disease. This volume provides a guided tour of the role that mathematical models have played in epidemiology and public health policy. This tour introduces a wide range of models and tools that have proven useful in the study of disease dynamics and control. This book provides a framework that will position those interested in the use of modeling and computational tools in epidemiology, public health, and related fields, in a position to contribute to the study of the transmission dynamics and control of contagion.

1.2 Some History The study of infectious disease data began with the work of John Graunt (1620– 1674) in his 1662 book “Natural and Political Observations made upon the Bills of Mortality.” The Bills of Mortality were weekly records of numbers and causes of death in London parishes. The records, beginning in 1592 and kept continuously from 1603 on, provided the data that Graunt used to begin to understand or identify possible causes of observed mortality patterns. He analyzed the various causes of death and gave a method of estimating the comparative risks of dying from various diseases, giving the first approach to a theory of competing risks. In the eighteenth century smallpox was endemic and, perhaps not surprisingly, the first model in mathematical epidemiology was tied in to the work that Daniel Bernoulli (1700–1782) carried out on estimating the impact of inoculation against smallpox. Variolation, essentially inoculation with a mild strain, was introduced as a way to produce lifelong immunity against smallpox, but with a small risk of infection and death. There was heated debate about variolation, and Bernoulli was led to study the question of whether variolation was beneficial. His approach was to calculate the increase in life expectancy if smallpox were to be eliminated as a cause of death. His approach to the question of competing risks led to the publication of

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a brief outline in 1760 [7] followed in 1766 by a more complete exposition [8]. His work received a mainly favorable reception; research that has become known in the actuarial literature rather than in the epidemiological literature. More recently his approach has been generalized [31]. Another valuable contribution to the understanding of infectious diseases prior to our understanding of disease transmission processes was gained from the study of the temporal and spatial pattern of cholera cases during the 1855 epidemic in London carried out by John Snow. He was able to pinpoint the Broad Street water pump as the source of the infection [54, 71]. In 1873, William Budd was able to gain a similar understanding of the spread of typhoid [17]. Statistical theory also moved forward with William Farr’s study of statistical returns in 1840, a study that had as its goal the discovery of the laws that underlie the rise and fall of epidemics [36]. Many of the early developments in the mathematical modeling of communicable diseases are due to public health physicians. The first known result in mathematical epidemiology, as noted before, is a defense of the practice of inoculation against smallpox in 1760 by Daniel Bernoulli, a member of a famous family of mathematicians (eight spread over three generations) who had been trained as a physician. The first contributions to modern mathematical epidemiology are due to P.D. En’ko between 1873 and 1894 [30], and the foundations of the entire approach to epidemiology based on compartmental models were laid by public health physicians such as Sir R.A. Ross, W.H. Hamer, A.G. McKendrick, and W.O. Kermack between 1900 and 1935, along with important contributions from a statistical perspective by J. Brownlee.

1.2.1 The Beginnings of Compartmental Models In order to describe a mathematical model for the spread of a communicable disease, it is necessary to make some assumptions about the means of spreading infection. The idea of invisible living creatures as agents of disease goes back at least to the writings of Aristotle (384–322 BC). The existence of microorganisms was demonstrated by van Leeuwenhoek (1632–1723) with the aid of the first microscopes. The first expression of the germ theory of disease by Jacob Henle (1809–1885) came in 1840 and was developed by Robert Koch (1843–1910), Joseph Lister (1827–1912), and Louis Pasteur (1822–1875) in the late nineteenth and early twentieth centuries. The modern view is that many diseases are spread by contact through a virus or bacterium. We focus in this book on the problem of understanding the spread of disease at a population level. Similar modeling approaches can be used to study the dynamics of infection within a host for diseases including HIV. This area is the backbone of the field of mathematical and computational immunology and viral dynamics. An introduction to immunology may be found in the book by Nowak and May [67]. In 1906, W.H. Hamer argued that the spread of infection should depend on the number of susceptible individuals and the number of infective individuals [44]. He

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suggested a mass action law for the rate of new infections, and this idea has been basic in the formulation of compartmental models since that time. It is worth noting that the foundations of the entire approach to epidemiology based on compartmental models were laid, not by mathematicians, but primarily by public health physicians such as Sir R.A. Ross, W.H. Hamer, A.G. McKendrick, and W.O. Kermack between 1900 and 1935. A particularly instructive example is the work of Ross on malaria. Sir Ronald Ross was awarded the second Nobel Prize in Medicine in 1902 for his demonstration of the dynamics of the transmission of malaria between mosquitoes and humans. He discovered the malarial parasite in the gastrointestinal tract of the Anopheles mosquito from which he was able to characterize the life cycle of malaria. He concluded that this vector-borne disease was transmitted by the Anopheles mosquito and in the process he developed a program for controlling or eliminating it at the population level. It was generally believed that, so long as mosquitoes were present in a population, malaria could not be eliminated. Ross introduced a simple compartmental model [69] that included mosquitoes and humans. He showed that reducing the mosquito population below a critical level would be sufficient to eliminate malaria. This was the first introduction of the concept of the basic reproduction number, a central idea in mathematical epidemiology since that time. Field trials supported Ross’ conclusion leading sometimes to brilliant successes in malaria control. The basic compartmental models to describe the transmission of communicable diseases are contained in a sequence of three papers by W.O. Kermack and A.G. McKendrick in 1927, 1932, and 1933 [55–57]. The first of these papers described epidemic models. The Kermack–McKendrick epidemic model, introduced in Chap. 2 and studied in more detail in Chap. 4, included dependence on age of infection, that is, the time since becoming infected, and can be used to provide a unified approach to compartmental epidemic models. Various disease outbreaks including the SARS epidemic of 2002–2003, the concern about a possible H 5N1 influenza epidemic in 2005, the H 1N1 influenza pandemic of 2009, and the Ebola outbreak of 2014 have reignited interest in epidemic models, with the reformulation of the Kermack–McKendrick model by Diekmann, Heesterbeek, and Metz [27] highlighting the importance of looking at the foundational work. Chapter 4 contains a study of epidemic models. In the work of Ross and Kermack and McKendrick there is a threshold quantity, the basic reproduction number, which is now almost universally denoted by R0 . Neither Ross nor Kermack and McKendrick identified this threshold quantity or gave it a name. It appears that the first person to name the threshold quantity explicitly was MacDonald [60] in his work on malaria. The basic reproduction number, R0 (referred to as the basic reproductive number by some authors), is defined as the expected number of disease cases (secondary infections) produced by a “typical” infected individual in a wholly susceptible population over the full course of the disease outbreak. In an epidemic situation, in which the time period is short enough to neglect demographic effects and all

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infected individuals recover with full immunity against reinfection, the threshold R0 = 1 is the dividing line between the infection dying out and the onset of an epidemic. In a situation that includes a flow of new susceptible individuals, either through demographic effects or recovery without full immunity against reinfection, the threshold R0 = 1 is the dividing line between an approach to a disease-free equilibrium and an approach to an endemic equilibrium, in which the disease is always present. This situation is studied in detail in Chap. 3. Since 1933, there has been a great deal of work on compartmental disease transmission models, with generalizations in many directions. In particular, it is assumed in [55–57] that stays in compartments are exponentially distributed. In the generalization to age of infection models in Chap. 4, we are able to assume arbitrary distributions of stay in a compartment.

1.2.2 Stochastic Models There are serious shortcomings in the simple Kermack–McKendrick model as a description of the beginning of a disease outbreak. Indeed, a very different kind of model is required since the Kermack–McKendrick compartmental epidemic model assumes that the sizes of the compartments are large enough that the mixing of members is homogeneous. However, at the beginning of a disease outbreak, there is a very small number of infective individuals and the transmission of infection is better captured if seen as a stochastic event that depends on the pattern of contacts between members of the population; a more satisfactory description should take such a stochastic pattern into account. We will not study stochastic models in this volume except for two sections at the start of Chap. 4 dealing with the initial stages of a disease outbreak. The process chosen here to describe it is known as a Galton–Watson process; the result was first given in [39, 77] although there was a gap in the convergence proof. The first complete proof was given much later by Steffensen [72, 73]. The result is now a standard theorem given in many sources on branching processes, for example, [45], but did not appear in the epidemiological literature until later. To the best of the authors’ knowledge, the first description in an epidemiological reference is [62] and the first epidemiological book source is the book by O. Diekmann and J.A.P. Heesterbeek [26] in 2000. A stochastic branching process description of the beginning of a disease outbreak begins with the assumption that there is a network of contacts of individuals, which may be described by a graph with members of the population represented by vertices and with contacts between individuals represented by edges. The study of graphs originated with the abstract theory of Erdös and Rényi of the 1950s and 1960s [33– 35], and has become important more recently in many areas, including in the study of social contacts, computer networks, and many other areas, as well as in the spread of communicable diseases. We will think of networks as bi-directional, with disease transmission possible in either direction along an edge. A brief taste of network

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models is given at the beginning of Chap. 4. It is however important to stress the fact that this book does not get involved in the study of stochastic models or in the study of epidemics in networks–areas that deserve their own volumes. We consider a disease outbreak that begins with a single infected individual (“patient zero”) who transmits infection to every individual to whom this individual is connected, that is, along every edge of the graph from the vertex corresponding to this individual. In other words, we assume that a disease outbreak begins when a single infective transmits infection to all of the people with whom he or she is in contact. Our development via branching processes is along the lines of that of [26]. Another approach, using a contact network perspective taken in [20, 65, 66] begins with an infected edge, corresponding to a disease outbreak started by an infective individual who passes the infection on to only one contact. This approach is the one taken more commonly in studies of epidemics on networks. It is somewhat more complicated and leads to somewhat different results, although the methods are quite similar. In a stochastic setting, it is possible to prove that there is also a number called the basic reproduction number denoted by R0 with the properties that if R0 < 1 the probability that the infection will die out is 1, while if R0 > 1 there is a positive probability that the infection will persist leading to an epidemic. However, there is also a positive probability that the infection will increase initially but will produce only a minor outbreak dying out before triggering a major epidemic. This distinction between a minor outbreak and a major epidemic, and the result that if R0 > 1 there may be only a minor outbreak and not a major epidemic is intrinsic in these stochastic models and is not reflected in deterministic models, the primary theme of this book. A possible approach to a realistic description of an epidemic would consider the use of a branching process model initially, making a transition to a compartmental model when the epidemic has become established, that is, when there are enough infectives so that the mass action mixing in the population becomes a reasonable approximation. Another approach would be to continue to use a network model throughout the course of the epidemic [63, 64, 76]. It is possible to formulate this model dynamically with the limiting case of this dynamic model, as the population size becomes very large, being the same as the compartmental model. Past experiences and data have shown that the spread of infection in small populations is better captured in small communities as a random process. For this reason, stochastic models have an important role in disease transmission modeling. The most commonly used stochastic model includes the chain binomial model of Reed and Frost, first described in lectures in 1928 by W.H. Frost but not published until much later [1, 78]. The Reed–Frost model was actually anticipated nearly 40 years earlier by P.D. En’ko [30]. The work of En’ko was brought to public attention much later by K. Dietz [28]. E.B. Wilson and M.H. Burke have given a description of Frost’s 1928 lectures with a somewhat different derivation [78]. M. Greenwood introduced a somewhat different chain binomial model in 1931 [40]. The Reed– Frost model has been used widely as a basic stochastic model and many extensions have been formulated. The book [25] by D.J. Daley and J. Gani contains an account

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of some of the more recent extensions. Also, a stochastic analogue of the Kermack– McKendrick epidemic model has been described in [6].

1.2.3 Developments in Compartmental Models In the mathematical modeling of disease transmission, as in most other areas of mathematical modeling, there is always a trade-off between simple, or strategic, models, which omit most details and are designed only to highlight general qualitative behavior, and detailed, or tactical, models, usually designed for specific situations including short-term quantitative predictions. Detailed models are generally difficult or impossible to solve analytically and hence their usefulness for theoretical purposes is limited, although their strategic value may be high. For example, very simple models for epidemics predict that an epidemic will die out after some time, leaving a part of the population untouched by disease, and this is also true of models that include control measures. This qualitative principle is not by itself very helpful in suggesting what control measures would be most effective in a given situation, but it implies that a detailed model describing the situation as accurately as possible might be useful for public health professionals. The ultimate in detailed models are agent-based models, which essentially divide the population into individuals or groups of individuals with identical behavior [32]. It is important to recognize that mathematical models to be used for making policy recommendations for management need quantitative results, and that the models needed in a public health setting require a great deal of detail in order to describe the situation accurately. For example, if the problem is to recommend what age group or groups should be the focus of attention in coping with a disease outbreak, it is essential to use a model which separates the population into a sufficient number of age groups and recognizes the interaction between different age groups. The development of high speed computing has made it possible to analyze highly detailed models rapidly. The development of mathematical methods for the study of models for communicable diseases led to a divergence between the goals of mathematicians, who sought broad understanding, and public health professionals, who sought practical procedures for management of diseases. While mathematical modeling led to many fundamental ideas, such as the possibility of controlling smallpox by vaccination and the management of malaria by controlling the vector (mosquito) population, the practical implementation was always more difficult than the predictions of simple models. Fortunately, in recent years there have been determined efforts to encourage better communication, so that public health professionals can better understand the situations in which simple models may be useful and mathematicians can recognize that real-life public health questions are much more complicated than simple models. In the study of compartmental disease transmission models, the population under study is divided into compartments and assumptions are made about the nature

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and time rate of transfer from one compartment to another. For example, in an SI R model, we divide the population being studied into three classes labeled S, I , and R. We let S(t) denote the number of individuals who are susceptible to the disease, that is, who are not (yet) infected at time t. I (t) denotes the number of infected individuals, assumed infectious and able to spread the disease by contact with susceptible individuals. R(t) denotes the number of individuals who have been infected and then removed from the possibility of being infected again or of spreading infection. In an SI S model, infectives recover with immunity against reinfection and the transitions are from susceptible to infective to susceptible. The rates of transfer between compartments are expressed mathematically as derivatives with respect to time of the sizes of the compartments. Initially, we assume that the duration of stay in each compartment is exponentially distributed, and as a result models are formulated initially as differential equations. Models in which the rates of transfer depend on the sizes of compartments over the past, as well as at the instant of transfer, lead to more general types of functional equations, such as differential–difference equations or integral equations. One way in which models have expressed the idea of a reduction in contacts as an epidemic proceeds is to assume a contact rate of the form βSf (I ) with a function f (I ) that grows more slowly than linearly in I . Such an assumption, while not really a mechanistic model, may give better approximation to observed data than mass action contact. In the simple Kermack–McKendrick epidemic model there are two parameters, the rate of new infections and the recovery rate. Often, the recovery rate for a particular disease is known. In compartmental epidemic models with more compartments, the progression rate is more complicated but may also be known. It is possible to estimate the basic reproduction number if these parameters can be estimated, and there is an equation, known as the final size relation relating the basic reproduction number to the number of individuals infected over the course of the epidemic. There have been many presentations of this final size relation in various contexts, including models with heterogeneity of mixing [3, 12, 13, 15, 16, 26, 27, 59]. In their later work on disease transmission models [56, 57], Kermack and McKendrick did not include age of infection, and age of infection models were neglected for many years. Age of infection reappeared in the study of HIV/AIDS, in which the infectivity of infected individuals is high for a brief period after becoming infected, then quite low for an extended period, possibly several years, before increasing rapidly with the onset of full-blown AIDS. Thus the age of infection described by Kermack and McKendrick for epidemics became very important in some endemic situations; see, for example, [74, 75]. Also, HIV/AIDS has pointed to the importance of immunological ideas in the analysis on the epidemiological level. Typically, there is a stochastic phase at the beginning of a disease outbreak, followed by an exponential increase in the number of infectives, and it may be possible to estimate this initial exponential growth rate experimentally. An estimate of the initial exponential growth rate can be used to estimate the rate of new infections, and this enables estimation of the basic reproduction number

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The development and analysis of compartmental models has grown rapidly since the early models. Many of these developments are due to H.W. Hethcote [46–50]. We describe only a few of the important developments. While there are three basic compartmental disease transmission models, namely the SI S model, the SI R model without births and deaths, and the SI R model with births and deaths, each disease has its own properties which should be included in a model. We will describe the effects of adding heterogeneity of mixing in several chapters, including heterogeneity of mixing in Chap. 5, age structure in Chap. 13, and spatial heterogeneity in Chaps. 14 and 15. In addition, the chapters on individual diseases (Chaps. 7–12) include modeling aspects of the specific disease being studied. For influenza, there is a significant fraction of the population which is infected but asymptomatic, with lower infectivity than symptomatic individuals. There are seasonal outbreaks which may be closely related to the strain of the previous year, that is, modified by point mutations, or it may be less closely related or it may be unrelated. The level of relation of two strains, as measured by the response of hosts that have experienced a prior influenza infection, is what we refer to as strain crossimmunity. Cross-immunity measures the level of protection earned by individuals who were infected by a related strain in a previous year. Influenza models may consider the effect that a partially efficacious vaccination may have before an outbreak, as well as the role of antiviral treatment during an outbreak. Considering multiple modes of transmission is critical when the goal is to minimize the impact of a disease in a population. Cholera may be transmitted both by direct contact and by contact with pathogen shed by infectives. In tuberculosis some individuals progress rapidly to active tuberculosis, while others progress much more slowly. Also, active-tuberculosis individuals who fail to comply with long-term treatment schedules become prime candidates for the development of a drug-resistant strain. In HIV/AIDS, the infectivity of an individual depends strongly on the time since infection, while transmission depends on modes of transmission, mixing between individuals and more. In malaria, immunity against infection is boosted by exposure to infection or by genetics (sickle cell anemia).

1.2.4 Endemic Disease Models The analytic approaches to models for endemic diseases and epidemics are quite different. The analysis of a model for an endemic disease, carried out in Chap. 3, begins with the search for equilibria, which are, by definition, constant solutions of the model. Usually there is a disease-free equilibrium and there are one or more endemic equilibria, with a positive number of infected individuals. The next step is to linearize about each equilibrium and determine the stability of each equilibrium. Usually, if the basic reproduction number is less than 1, the only equilibrium is the disease-free equilibrium and this equilibrium is asymptotically stable. If the basic reproduction number is greater than 1, the usual situation is that the diseasefree equilibrium is unstable and there is a unique endemic equilibrium which is

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asymptotically stable. This approach also covers diseases in which there is vertical transmission, which is direct transmission from mother to offspring at birth [19]. However, more complicated behavior is possible. For example, if there are two strains of the disease being studied it is common to have regions in the parameter space in which there is an asymptotically stable equilibrium with only one of the strains present and a region in which there is an asymptotically stable equilibrium with both strains coexisting. Another possibility is that there is a unique endemic equilibrium but it is unstable. In this situation, there is often a Hopf bifurcation and an asymptotically stable periodic orbit around the endemic equilibrium. An example of such behavior may be found in an SIRS model, with a temporary immunity period of fixed length following recovery [51] and in an SV I R model [38]. If there is a periodic orbit with large amplitude and a long period, data must be gathered over a sufficiently large time interval to give an accurate picture. Another possible behavior is a backward bifurcation. As R0 increases through 1 there is an exchange of stability between the disease-free equilibrium, which is asymptotically stable for R0 < 1 and unstable for R0 > 1, and the endemic equilibrium which exists if R0 > 1. The usual transition is a forward, or transcritical, bifurcation at R0 = 1, with an asymptotically stable endemic equilibrium and an equilibrium infective population size depending continuously on R0 . The behavior at a bifurcation may be described graphically by the bifurcation curve, which is the graph of the infective population size I at equilibrium as a function of the basic reproduction number R0 . It has been noted [29, 42, 43, 58] that in epidemic models with multiple groups and asymmetry between groups or multiple interaction mechanisms it is possible to have a very different bifurcation behavior at R0 = 1. There may be multiple positive endemic equilibria for values of R0 < 1 and a backward bifurcation at R0 = 1. The qualitative behavior of a system with a backward bifurcation differs from that of a system with a forward bifurcation and the nature of these changes has been described in [11]. Since these behavioral differences are important in planning how to control a disease, it is important to determine whether a system can have a backward bifurcation. In the presence of two modes of sexually transmitted HIV, it was shown that multiple endemic equilibrium could be supported [53].

1.2.5 Diseases Transmitted by Vectors Many diseases are transmitted from human to human indirectly, through a vector. Vectors are living organisms that can transmit infectious diseases between humans. Many vectors are bloodsucking insects that ingest disease-producing microorganisms during blood meals from an infected (human) host, and then inject it into a new host during a subsequent blood meal. The best known vectors are mosquitoes for diseases including malaria, dengue fever, chikungunya, Zika virus, Rift Valley fever, yellow fever, Japanese encephalitis, lymphatic filariasis, and West Nile fever, but ticks (for Lyme disease and tularemia), bugs (for Chagas’ disease), flies (for

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onchocerciasis), sandflies (for leishmaniasis), fleas (for plague, transmitted by fleas from rats to humans), and some freshwater snails (for schistosomiasis) are vectors for some diseases. Every year there are more than a billion cases of vector-borne diseases and more than a million deaths. Vector-borne diseases account for over 17% of all infectious diseases worldwide. Malaria is the most deadly vector-borne diseases, causing an estimated 627,000 deaths in 2012. The most rapidly growing vector-borne disease is dengue, for which the number of cases has multiplied by 30 in the last 50 years. These diseases are found more commonly in tropical and sub-tropical regions where mosquitoes flourish, and in places where access to safe drinking water and sanitation systems is uncertain. Some vector-borne diseases such as dengue, chikungunya, and West Nile virus are emerging in countries where they were unknown previously because of globalization of travel and trade and environmental challenges such as climate change. A troubling new development is the Zika virus, which has been known since 1952 but has developed a mutation in the South American outbreak of 2015 [70] which has produced very serious birth defects in babies born to infected mothers. In addition, the current Zika virus can be transmitted directly through sexual contact as well as through vectors. Chapter 6 is an introduction to the modeling of vector-borne diseases. Chapter 11 on malaria and Chap. 12 on dengue fever and the Zika virus describe modeling of specific vector-borne diseases. Many of the important underlying ideas of mathematical epidemiology arose in the study of malaria begun by Sir R.A. Ross [69]. Malaria is one example of a disease with vector transmission, the infection being transmitted back and forth between vectors (mosquitoes) and hosts (humans). It kills hundreds of thousands of people annually, mostly children and mostly in poor countries in Africa. Among communicable diseases, only tuberculosis causes more deaths. Other vector diseases include West Nile virus, yellow fever, and dengue fever. Human diseases transmitted heterosexually may also be viewed as diseases transmitted by vectors, because males and females must be viewed as separate populations and disease is transmitted from one population to the other. Vector-transmitted diseases require models that include both vectors and hosts. For most diseases transmitted by vectors, the vectors are insects, with a much shorter life span than the hosts, who may be humans as for malaria or animals as for West Nile virus, although there is malaria (not human malaria) in various animal populations and West Nile virus has infected humans as far as Arizona in the USA. The compartmental structure of the disease may be different in host and vector species; for many diseases with insects as vectors an infected vector remains infected for life so that the disease may have an SI or SEI structure in the vectors and an SI R or SEI R structure in the hosts.

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1.2.6 Heterogeneity of Mixing In disease transmission models not all members of the population make contacts at the same rate. In sexually transmitted diseases there is often a “core” group of very active members who are responsible for most of the disease cases, and control measures aimed at this core group have been very effective in control [52]. In epidemics there are often “super-spreaders,” who make many contacts and are instrumental in spreading disease and in general some members of the population make more contacts than others. Chapter 5 deals with models for diseases with heterogeneous mixing and includes description of a general method (the next generation matrix) for determining the basic reproduction number for models with heterogeneous mixing. To model heterogeneity in mixing we may assume that the population is divided into subgroups with different activity levels. Formulation of models requires some assumptions about the mixing between subgroups. There have been many studies of mixing patterns in real populations, for example, [9, 10, 18, 37, 68]. It has often been observed in epidemics that most infectives do not transmit infections at all or transmit infections to very few others. This suggests that homogeneous mixing at the beginning of an epidemic may not be a good approximation. The SARS epidemic of 2002–2003 spread much more slowly than would have been expected on the basis of the data on disease spread at the start of the epidemic. Early in the SARS epidemic of 2002–2003 it was estimated that R0 had a value between 2.2 and 3.6. At the beginning of an epidemic, the exponential rate of growth of the number of infectives is approximately (R0 −1)/α, where 1/α is the generation time of the epidemic, estimated to be approximately 10 days for SARS. This would have predicted at least 30,000 cases of SARS in China during the first 4 months of the epidemic. In fact, there were fewer than 800 cases reported in this time. One explanation for this discrepancy is that the estimates were based on transmission data in hospitals and crowded apartment complexes or in the scaling of the model used to estimate parameters [24]. It was observed that there was intense activity in some locations and very little in others. This suggests that the actual reproduction number (averaged over the whole population) was much lower, perhaps in the range 1.2–1.6, and that heterogeneous mixing was a very important aspect of the epidemic. Age is one of the most important characteristics in the modeling of populations and infectious diseases. Individuals with different ages may have different reproduction and survival capacities. Diseases may have different infection rates and mortality rates for different age groups. Individuals of different ages may also have different behaviors, and behavioral changes are crucial in control and prevention of many infectious diseases. Young individuals tend to be more active in interactions with or between populations, and in disease transmissions. Age-structured models are studied in Chap. 13. Sexually transmitted diseases (STDs) are spread through partner interactions with pair-formations and the pair-formations process is age-dependent in most cases. For example, most HIV cases occur in the group of young adults.

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Childhood diseases, such as measles, chicken pox, and rubella, are spread mainly by contacts between children of similar ages. More than half of the deaths attributed to malaria are in children under 5 years of age due to their weaker immune systems. This suggests that in models for disease transmission in an age-structured population it is necessary to allow the contact rates between two members of the population to depend on the ages of both members. Another important motivation for using agestructured models for childhood diseases is that vaccination is age-dependent (e.g., measles). The development of age-structured models for disease transmission required development of the theory of age-structured populations. In fact, the first models for age-structured populations [61] were designed for the study of disease transmission in such populations.

1.3 Strategic Models and This Volume This book is intended for both mathematicians and public health professionals. However, it is a book aimed primarily at practitioners who are quantitatively trained. For readers less familiar with mathematics, we are providing a website that includes notes on calculus, linear algebra, ODEs, and difference equations. We would like to repeat that it is not a book about mathematics per se. Hence, mathematical rigor is not a priority albeit we have tried to document some assertions. We wish to present the truth and nothing but the truth, but not necessarily the whole truth. Our hope is that public health people will be sufficiently interested to review mathematics if necessary and make use of the book. We do not cover stochastic models, except for some material in Sects. 4.1 and 4.2. Some presentations of stochastic models may be found in [2, 4, 5, 25, 41]. Also, we do not cover discrete models in the main text. This is a deliberate choice that we have made to limit the size of this volume, despite the fact that a lot can be learned by the appropriate formulation of discrete models, which are unfortunately often seen as a discretization of continuous time models, that is, they are not derived directly from first principles, deviating the focus from the study of disease dynamics to the mathematical question of whether or not it is a proper discretization of a continuous time model. However, there are several projects that could give an introduction to this topic, and some references are [14, 21–23, 79]. The primary goal of this volume is to cover relatively simple models to indicate what qualitative results should be expected from more detailed tactical models. The chapters build on the work that we have done in applying epidemiological models in the context of specific diseases over the past 25 years—including some of our most recent work. It is our hope that the questions and problems highlighted in this introduction may help build collaborations between modelers, epidemiologists, and public health experts.

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References 1. Abbey, H. (1952) An estimation of the Reed–Frost theory of epidemics, Human Biol. 24; 201– 223. 2. Allen, L.J.S. (2003) An Introduction to Stochastic Processes with Applications to Biology, Pearson Education Inc. 3. Andreasen, V. (2011) The final size of an epidemic and its relation to the basic reproduction number, Bull. Math. Biol. 73: 2305–2321. 4. Bailey. N.T.J. (1975) The Mathematical Theory of Infectious Diseases and its Applications, Oxford Univ. Press. 5. Ball, F.G. (1983) The threshold behaviour of epidemic models, J. App. Prob. 20: 227–241. 6. Bartlett, M.S. (1949) Some evolutionary stochastic processes J. Roy. Stat. Soc. B 11: 211–229. 7. Bernoulli, D. (1760) Réflexions sur les avantages de l’inoculation, Mercure de Paris, p. 173. 8. Bernoulli, D. (1766) Essai d’une nouvelle analyse de la mortalité causeé par la petite vérole, Mem. Math. Phys. Acad. Roy. Sci. Paris, 1–45. 9. Blythe, S.P., S. Busenberg & C. Castillo-Chavez (1995) Affinity and paired-event probability, Math. Biosc. 128: 265–84 . 10. Blythe, S.P., C. Castillo-Chavez, J. Palmer & M. Cheng (1991) Towards a unified theory of mixing and pair formation, Math. Biosc. 107: 379–405. 11. Brauer, F. (2004) Backward bifurcations in simple vaccination models, J. Math. Anal. & Appl. 298: 418–431. 12. Brauer, F. (2008) Epidemic models with treatment and heterogeneous mixing, Bull. Math. Biol. 70: 1869–1885. 13. Brauer, F. (2008) Age of infection models and the final size relation, Math. Biosc. & Eng. 5: 681–690. 14. Brauer, F., C. Castillo-Chavez, and Z. Feng (2010) Discrete epidemic models, Math. Biosc. & Eng. 7: 1–15. 15. Brauer, F. & J. Watmough (2009) Age of infection models with heterogeneous mixing, J. Biol. Dyn. 3: 324–330. 16. Breda, D., O. Diekmann, W.F. deGraaf, A. Pugliese, & R. Vermiglio (2012) On the formulation of epidemic models (an appraisal of Kermack and McKendrick, J. Biol. Dyn. 6: 103–117. 17. Budd, W. (1873) Typhoid Fever; Its Nature, Mode of Spreading, and Prevention, Longmans, London. 18. Busenberg, S. & C. Castillo-Chavez (1989) Interaction, pair formation and force of infection terms in sexually transmitted diseases, In Mathematical and Statistical Approaches to AIDS Epidemiology, Lect. Notes Biomath. 83, C. Castillo-Chavez (ed.), Springer-Verlag, BerlinHeidelberg-New York, pp. 289–300. 19. Busenberg, S. & K.L. Cooke (1993) Vertically Transmitted Diseases: Models and Dynamics, Biomathematics 23, Springer-Verlag, Berlin-Heidelberg-New York. 20. Callaway, D.S., M.E.J. Newman, S.H. Strogatz, & D.J. Watts (2000) Network robustness and fragility: Percolation on random graphs, Phys. Rev. Letters, 85: 5468–5471. 21. Castillo-Chavez, C. & A. Yakubu (2001) Discrete-time SI S models with simple and complex dynamics, Nonlinear Analysis, Theory. Methods, and Applications 47: 4753–4762. 22. Castillo-Chavez, C. & A. Yakubu (2002) Discrete-time SI S models with simple and complex population dynamics, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases IMA Vol. 125; 153–164. 23. Castillo-Chavez, C. & A. Yakubu (2002) Intraspecific competition, dispersal and disease dynamics in discrete-time patchy environments, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases IMA Vol. 125; 165–181. 24. Chowell, G., P.W. Fenimore, M.A. Castillo-Garsow, and C.Castillo-Chavez (2003) SARS outbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation as a control mechanism. J. Theor. Biol. 224: 1–8.

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25. Daley, D.J. & J. Gani (1999) Epidemic Models: An Introduction, Cambridge Studies in Mathematical Biology 15, Cambridge University Press. 26. Diekmann, O. & J.A.P. Heesterbeek (2000) Mathematical epidemiology of infectious diseases: Model building, analysis and interpretation, John Wiley and Sons, ltd. 27. Diekmann,O., Heesterbeek,J.A.P., Metz,J.A.J. (1995) The legacy of Kermack and McKendrick, in Mollison,D. (ed)Epidemic Models: Their Structure and Relation to Data. Cambridge University Press, Cambridge, pp. 95–115. 28. Dietz, K, (1988) The first epidemic model: a historical note on P.D. En’ko, Australian J. Stat. 30A: 56–65. 29. Dushoff, J., W. Huang, & C. Castillo-Chavez (1998) Backwards bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol. 36: 227–248. 30. En’ko, P.D. (1889) On the course of epidemics of some infectious diseases, Vrach. St. Petersburg, X: 1008–1010, 1039–1042, 1061–1063. [translated from Russian by K. Dietz, Int. J. Epidemiology (1989) 18: 749–755]. 31. Dietz, K. & J.A.P. Heesterbeek (2002) Daniel Bernoulli’s epidemiological model revisited, Math. Biosc. 180: 1–221. 32. Epstein, J.M. & R.L. Axtell (1996) Growing Artificial Societies: Social Science from the Bottom Up, MIT/Brookings Institution. 33. Erdös, P. & A. Rényi (1959) On random graphs, Publicationes Mathematicae 6: 290–297. 34. Erdös, P. & A. Rényi (1960) On the evolution of random graphs, Pub. Math. Inst. Hung. Acad. Science 5: 17–61. 35. Erdös, P. & A. Rényi (1961) On the strengths of connectedness of a random graph, Acta Math. Scientiae Hung. 12: 261–267. 36. Farr, W. (1840) Progress of epidemics, Second Report of the Registrar General of England and Wales, 91–98. 37. Feng, Z., Hill, A.N., Smith, P.J. Glasser, J.W. (2015) An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing, J. Theor. Biol. 386: 177–187. 38. Feng, Z. and H.R. Thieme (1995) Recurrent outbreaks of childhood diseases revisited: the impact of isolation, Math. Biosc. 128: 93–130. 39. Galton, F. (1889) Natural Inheritance (2 nd ed.), App. F, pp. 241–248. 40. Greenwood, M. (1931) On the statistical measure of infectiousness, J. Hygiene 31: 336–351. 41. Greenwood, P.E. & L.F. Gordillo (2009) Stochastic epidemic modeling, in Mathematical and Statistical Estimation Approaches in Epidemiology, G. Chowell, J.M. Hyman, L.M.A. Bettencourt, & C. Castillo-Chavez (eds.), Springer, pp. 31–52. 42. Hadeler, K.P. & C. Castillo-Chavez (1995) A core group model for disease transmission, Math Biosc. 128: 41–55. 43. Hadeler, K.P. & P. van den Driessche (1997) Backward bifurcation in epidemic control, Math. Biosc. 146: 15–35. 44. Hamer, W.H. (1906) Epidemic disease in England - the evidence of variability and of persistence, The Lancet 167: 733–738. 45. Harris, T.E. (1963) The Theory of Branching Processes, Springer. 46. Hethcote, H.W. (1976) Qualitative analysis for communicable disease models, Math. Biosc., 28:335–356. 47. Hethcote, H.W. (1978) An immunization model for a heterogeneous population, Theor. Pop. Biol., 14:338–349. 48. Hethcote, H.W. (1989) Three basic epidemiological models. In Applied Mathematical Ecology, S. A. Levin, T.G. Hallam and L.J. Gross, eds., Biomathematics 18: 119–144, Springer-Verlag, Berlin-Heidelberg-New York. 49. Hethcote, H.W. (1997) An age-structured model for pertussis transmission, Math. Biosci., 145: 89–136. 50. Hethcote, H.W. (2000) The mathematics of infectious diseases, SIAM Review, 42: 599–653. 51. Hethcote, H.W., H.W. Stech and P. van den Driessche (1981) Periodicity and stability in epidemic models: a survey. In: S. Busenberg & K.L. Cooke (eds.) Differential Equations and

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Applications in Ecology, Epidemics and Population Problems, Academic Press, New York: 65–82. 52. Hethcote, H.W. & J.A. Yorke (1984) Gonorrhea Transmission Dynamics and Control, Lect. Notes in Biomath. 56, Springer-Verlag. 53. Huang, W., K.L. Cooke, & C. Castillo-Chavez (1992) Stability and bifurcation for a multiple group model for the dynamics of HIV/AIDS transmission, SIAM J. App. Math. 52: 835–853. 54. Johnson, S. (2006) The Ghost Map, Riverhead Books, New York, 55. Kermack, W.O. & A.G. McKendrick (1927) A contribution to the mathematical theory of epidemics, Proc. Royal Soc. London, 115:700–721. 56. Kermack, W.O. & A.G. McKendrick (1932) Contributions to the mathematical theory of epidemics, part. II, Proc. Roy. Soc. London, 138:55–83. 57. Kermack, W.O. & A.G. McKendrick (1933) Contributions to the mathematical theory of epidemics, part. III, Proc. Roy. Soc. London, 141:94–112. 58. Kribs-Zaleta, C.K. & J.X. Velasco-Hernandez (2000) A simple vaccination model with multiple endemic states, Math Biosc. 164: 183–201. 59. Ma, J. & D.J.D. Earn (2006) Generality of the final size formula for an epidemic of a newly invading infectious disease, Bull. Math. Biol. 68: 679–702. 60. MacDonald, G. (1957) The Epidemiology and Control of Malaria, Oxford University Press. Oxford University Press. 61. McKendrick, A.G. (1926) Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc. 44: 98–130. 62. Metz, J.A.J. (1978) The epidemic in a closed population with all susceptibles equally vulnerable; some results for large susceptible populations and small initial infections, Acta Biotheoretica 78: 75–123. 63. Miller, J.C. (2011) A note on a paper by Erik Volz: SIR dynamics in random networks, J. Math. Biol. 62: 349–358. 64. Miller, J.C., & E. Volz (2013) Incorporating disease and population structure into models of SIR disease in contact networks, PLoS One 8: e69162, https://doi.org/10.1371/journal.pone. 0069162. 65. Newman, M.E.J. (2002) The spread of epidemic disease on networks, Phys. Rev. E 66, 016128. ks, Phys. Rev. E 66, 016128. 66. Newman, M.E.J., S.H. Strogatz, & D.J. Watts (2001) Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E 64, 026118. 67. Nowak, M., & R.M. May (1996) Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford Univ. Press. 68. Nold, A. (1980) Heterogeneity in disease transmission modeling, Math. Biosc. 52: 227–240. 69. Ross, R.A.(1911) The prevention of malaria (2nd edition, with Addendum). John Murray, London. 70. Schuler-Faccini, L. (2016) Possible association between Zika virus infections and microcephaly Brazil 2015, MMWR Morbidity and Mortality weekly report: 65. 71. Snow, J. (1855) The mode of communication of cholera (2 nd ed.), Churchill, London. 72. Steffensen, G.F. (1930) Om sandsynligheden for at affkommet uddor, Matematisk Tiddskrift B 1: 19–23. 73. Steffensen, G.F. (1931) Deux problèmes du calcul des probabilités, Ann. Inst. H. Poincaré 3: 319–341. 74. Thieme, H.R. and C. Castillo-Chavez (1989) On the role of variable infectivity in the dynamics of the human immunodeficiency virus. In Mathematical and statistical approaches to AIDS epidemiology, C. Castillo-Chavez, ed., Lect. Notes Biomath. 83, 200–217, Springer-Verlag, Berlin-Heidelberg-New York. 75. Thieme, H.R. and C. Castillo-Chavez (1993) How may infection-age dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53:1447–1479.

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76. Volz, E. (2008) SIR dynamics in random networks with heterogeneous connectivity, J. Math. Biol., 56: 293–310. 77. Watson, H.W. & F. Galton (1874) On the probability of the extinction of families, J. Anthrop. Inst. Great Britain and Ireland 4: 138–144. 78. Wilson, E.B. & M.H. Burke (1942) The epidemic curve, Proc. Nat. Acad. Sci. 28: 361–367. 79. Yakubu, A. & C. Castillo-Chavez (2002) Interplay between local dynamics and dispersal in discrete-time metapopulation models, J. Theor. Biol. 218: 273–288.

Chapter 2

Simple Compartmental Models for Disease Transmission

Communicable diseases that are endemic (always present in a population) cause many deaths. For example, in 2011 tuberculosis caused an estimated 1,400,000 deaths and HIV/AIDS caused an estimated 1,200,000 deaths worldwide. According to the World Health Organization there were 627,000 deaths caused by malaria, but other estimates put the number of malaria deaths at 1,2000,000. Measles, which is easily treated in the developed world, caused 160,000 deaths in 2011, but in 1980 there were 2,600,000 measles deaths. The striking reduction in measles deaths is due to the availability of a measles vaccine. Other diseases such as typhus, cholera, schistosomiasis, and sleeping sickness are endemic in many parts of the world. The effects of high disease mortality on mean life span and of disease debilitation and mortality on the economy in afflicted countries are considerable. Most of these disease deaths are in less developed countries, especially in Africa, where endemic diseases are a huge barrier to development. A reference describing the properties of many endemic diseases is [2]. For diseases that are endemic in some region, public health physicians would like to be able to estimate the number of infectives at a given time as well as the rate at which new infections arise. The effects of quarantine or vaccine in reducing the number of victims are of importance. In addition, the possibility of defeating the endemic nature of the disease and thus controlling or even eradicating the disease in a population is worthy of study. An epidemic, which acts on a short temporal scale, may be described as a sudden outbreak of a disease that infects a substantial portion of the population in a region before disappearing. Epidemics invariably leave part of the population untouched. Often, epidemic outbreaks recur with intervals of several years between outbreaks, possibly decreasing in severity as populations develop some immunity. The “Spanish” flu epidemic of 1919–1920 caused an estimated 50,000,000 or more deaths worldwide. The AIDS epidemic, the SARS epidemic of 2002–2003, recurring influenza pandemics such as the H1N1 influenza pandemic of 2009–2010,

© Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_2

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and outbreaks of diseases such as the Ebola virus are events of concern and interest to many people. The essential difference between an endemic disease and an epidemic is that in an endemic disease there is some mechanism for a flow of susceptibles into the population being studied through births of new susceptibles, immigration of susceptibles, recovery from infection without immunity against reinfection, or loss of immunity of recovered individuals. This may result in a level of infection that remains in the population. In an epidemic, there is no flow of new susceptibles into the population and the number of infectives decreases to zero because of the resulting scarcity of susceptibles. There are many questions of interest to public health physicians confronted with a possible epidemic. For example, how severe will an epidemic be? This question may be interpreted in a variety of ways. For example, how many individuals will be affected and require treatment? What is the maximum number of people needing care at any particular time? How long will the epidemic last? Can an epidemic be averted by vaccination of enough members of the population in advance of the epidemic? How much good would quarantine of victims do in reducing the severity of the epidemic? The idea of invisible living creatures as agents of disease goes back at least to the writings of Aristotle (384–322 BC) and was developed as a theory in the sixteenth century. The existence of microorganisms was demonstrated by Leeuwenhoek (1632–1723) with the aid of the first microscopes. The first expression of the germ theory of disease by Jacob Henle (1809–1885) came in 1840 and was developed by Robert Koch (1843–1910), Joseph Lister (1827–1912), and Louis Pasteur (1827– 1875) in the latter part of the nineteenth century and the early part of the twentieth century. The mechanism of transmission of infections is now known for most diseases. Generally, diseases transmitted by viral agents, such as influenza, measles, rubella (German measles), and chicken pox, confer immunity against reinfection, while diseases transmitted by bacteria, such as tuberculosis, meningitis, and gonorrhea, confer no immunity against reinfection. Other diseases, such as malaria, are transmitted not directly from human to human but by vectors, agents (usually insects) who are infected by humans and who then transmit the disease to humans. Heterosexual transmission of HIV/AIDS is also a vector process in which transmission goes back and forth between males and females. In this chapter, our goal is to present and analyze simple compartmental models for disease transmission, both for endemic situations and for epidemics. Here we seek only to introduce the idea of the basic reproduction number and how it determines a threshold between two different outcomes. In later chapters we will study models with more detailed structure and will include the effects of different methods to try to control a disease.

2.1 Introduction to Compartmental Models

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2.1 Introduction to Compartmental Models We formulate our descriptions of disease transmission as compartmental models, with the population under study being divided into compartments and with assumptions about the nature and time rate of transfer from one compartment to another. We divide the population being studied into three classes labeled S, I , and R. Let S(t) denote the number of individuals who are susceptible to the disease, that is, who are not (yet) infected at time t. I (t) denotes the number of infected individuals, assumed infective and able to spread the disease by contact with susceptibles. R(t) denotes the number of individuals who have been infected and then removed from the possibility of being infected again or of spreading infection. Removal is carried out either through isolation from the rest of the population, or through immunization against infection, or through recovery from the disease with full immunity against reinfection, or through death caused by the disease. These characterizations of removed members are different from an epidemiological perspective but are often equivalent from a modeling point of view which takes into account only the state of an individual with respect to the disease. In many diseases, infectives return to the susceptible class on recovery because the disease confers no immunity against reinfection. Such models are appropriate for most diseases transmitted by bacterial agents or helminth agents, and most sexually transmitted diseases (including gonorrhea, but not such diseases as AIDS from which there is no recovery). We use the terminology SI S to describe a disease with no immunity against reinfection, to indicate that the passage of individuals is from the susceptible class to the infective class and then back to the susceptible class. We will use the terminology SI R to describe a disease which confers immunity against reinfection, to indicate that the passage of individuals is from the susceptible class S to the infective class I to the removed class R. Usually, diseases caused by a virus are of SI R type. In addition to the basic distinction between diseases for which recovery confers immunity against reinfection and diseases for which recovered members are susceptible to reinfection, and the intermediate possibility of temporary immunity signified by a model of SI RS type, more complicated compartmental structure is possible. For example, there are SEI R and SEI S models, with an exposed period between being infected and becoming infective. We are assuming that the disease transmission process is deterministic, that is, that the behavior of a population is determined completely by its history and by the rules which describe the model. In formulating models in terms of the derivatives of the sizes of each compartment, we are also assuming that the number of members in a compartment is a differentiable function of time. This assumption is plausible in describing an endemic state. However, in Chap. 4 we will describe the modeling of disease outbreaks and epidemics. At the beginning of a disease outbreak, there are only a few infectives and the start of a disease outbreak depends on random contacts with a small number of infectives. This will require a different approach

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that we shall study in Sect. 4.1, but for the present we begin with a description of deterministic compartmental models. The independent variable in our compartmental models is the time t, and the rates of transfer between compartments are expressed mathematically as derivatives with respect to time of the sizes of the compartments, and as a result our models are formulated initially as differential equations. Models in which the rates of transfer depend on the sizes of compartments over the past as well as at the instant of transfer lead to more general types of functional equations, such as differential–difference equations or integral equations. In this chapter, we will always assume that the duration of stay in each compartment is exponentially distributed so that our models will be systems of ordinary differential equations. In Chaps. 3 and 4 we will begin the study of more general classes of models. In order to formulate a compartmental disease transmission model, we will need to make assumptions on the rates of flow between compartments. In the simplest models we require expressions for the rate of being infected and the rate of recovery from infection. The most common assumption about transmission of infection is mass action incidence. It is assumed that in a population of size N on average an individual makes βN contacts sufficient to transmit infection in unit time. Another possible assumption is standard incidence, in which it is assumed that on average an individual makes a constant number a of contacts sufficient to transmit infection in unit time. Standard incidence is a common assumption for sexually transmitted diseases. More realistically, one might assume that the parameters β or a are not constants, but are functions of total population size. We will always assume that contacts are effective. By this we mean that if there is a contact between an infective individual and a susceptible individual, infection is always passed to the susceptibles. However, it is possible that infectives in one compartment are less infectious than individuals in another compartment. In such situations, we may include a reduction factor in some contacts to model this. With mass action incidence, (contact rate βN ) since the probability that a random contact by an infective is with a susceptible is S/N, the number of new infections in unit time per infective is (βN)(S/N), giving a rate of new infections (βN)(S/N)I = βSI . Alternately, we may argue that for a contact by a susceptible the probability that this contact is with an infective is I /N and thus the rate of new infections per susceptible is (βN)(I /N), giving a rate of new infections (βN)(I /N)S = βSI . Note that both approaches give the same rate of new infections; in models with more complicated compartmental structure one approach may be more appropriate than the other. With standard incidence, (contact rate a) similar reasoning leads to the result that the rate of new infections is aS

I . N

If the total population size N is constant, we may use a and βN interchangeably. We will most commonly use βN , partly for historical reasons. However, for sexually

2.1 Introduction to Compartmental Models

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transmitted diseases the standard incidence form for contacts is more common. When we study heterogeneous mixing in Chap. 5 it will be more convenient to use the standard incidence form but it would be possible to use the mass action form instead. A common assumption is that infectives leave the infective class at a constant rate αI per unit time. This assumption requires a fuller mathematical explanation, since the assumption of a recovery rate proportional to the number of infectives has no clear epidemiological meaning. We consider the “cohort” of members who were all infected at one time and let u(s) denote the number of these who are still infective s time units after having been infected. If a fraction α of these leave the infective class in unit time then u′ = −αu, and the solution of this elementary differential equation is u(s) = u(0) e−αs . Thus, the fraction of infectives remaining infective s time units after having become infective is e−αs ,so that the length of the infective period is distributed ∞ exponentially with mean 0 e−αs ds = 1/α, and this is what is really assumed. This assumption is made for simplicity as it leads to ordinary differential equation models, but in a later chapter we will study models with other distributions of the infective period. To see that the mean infective period is 1/α, we point out since the fraction of infectives remaining infective s time units after being infected is e−αs , the fraction with infective period exactly s is −

d −αs e ds

and the mean infective period is 

∞

s

0

d −αs e ds. ds

Integration by parts shows that this is −



∞ 0

e−αs ds =

1 . α

In this chapter, we will analyze SI S and SI R models for disease transmission, both with and without demographics (births and natural deaths). Our goal is to describe both endemic and epidemic situations in the simplest possible contexts. In later chapters, we will analyze more complicated models, including more

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compartmental structure, more general distributions of stay in compartments, and heterogeneity of mixing. In this chapter we focus on equilibrium analysis and asymptotic behavior of models for endemic diseases and on final size relations for epidemic models. A constant solution y0 of a differential equation y ′ = f (y),

(2.1)

meaning a solution of the equation f (y) = 0, is called an equilibrium of the differential equation. An equilibrium y0 is said to be stable if every solution y(t) of the differential equation (2.1) with initial value y(0) sufficiently close to y0 remains close to the equilibrium y0 for all t ≥ 0. An equilibrium is said to be asymptotically stable if it is stable and if in addition every solution with initial value sufficiently close to y0 approaches the equilibrium as t → ∞. We note that in this definition, there is no specification of the meaning of “sufficiently close.” Asymptotic stability is a local concept. An equilibrium y0 is said to be globally asymptotically stable if it is stable and if solutions of (2.1) for all initial values y(0) approach y0 as t → ∞. An equilibrium that is not stable is said to be unstable. The concepts of asymptotic stability and instability are essential for the qualitative analysis of differential equations. An important basic result is that an equilibrium y0 of a differential equation (2.1) is asymptotically stable if f ′ (y0 ) < 0 and unstable if f ′ (y0 ) > 0. The case f ′ (y0 ) = 0 is more difficult to analyze. A central concept in both endemic and epidemic models is the basic reproduction number, to be defined in Sect. 2.2. In the next two sections, we will discuss models for endemic situations, and in the following two sections we will discuss epidemic models.

2.2 The SI S Model The basic compartmental models to describe the transmission of communicable diseases are contained in a sequence of three papers by W.O. Kermack and A.G. McKendrick in 1927, 1932, and 1933 [17–19]. The simplest SI S model, due to Kermack and McKendrick [18], is S ′ = −βSI + αI I ′ = βSI − αI.

(2.2)

It is based on the following assumptions: (i) The rate of new infections is given by mass action incidence. (ii) Infectives leave the infective class at rate αI per unit time and return to the susceptible class.

2.2 The SI S Model

27

(iii) There is no entry into or departure from the population. (iv) There are no disease deaths, and the total population size is a constant N. In an SI S model, the total population size N is equal to S + I . Later, we will allow the possibility that some infectives recover while others die of the disease to give a more general model. The hypothesis (iii) really says that the time scale of the disease is much faster than the time scale of births and deaths so that demographic effects on the population may be ignored. Because (2.2) implies (S + I )′ = 0, the total population N = S + I is a constant. We may reduce the model (2.2) to a single differential equation by replacing S by N −I 2 I ′ = βI (N − I )  − αI = (βN  − α)I − βI

= (βN − α)I 1 −

I N − βα

.

(2.3)

Now (2.3) is a logistic differential equation of the form   I , I ′ = rI 1 − K with r = βN − α and with K = N − α/β. Let us recall the analysis of the logistic equation. The logistic equation may be solved explicitly using separation of variables; the solution satisfying the initial condition x(0) = x0 is x(t) =

Kx0 ert Kx0 = . rt K − x0 + x0 e x0 + (K − x0 )e−rt

(2.4)

The expression (2.4) for the solution of the logistic initial value problem shows that x(t) approaches the limit K as t → ∞ if x0 > 0 provided r > 0, K > 0. If r < 0, K < 0, x(t) approaches the limit 0 as t → ∞. This case, K < 0 arises in the analysis of the differential equation (2.3), but does not have any biological meaning in itself. This qualitative result tells us that for the model (2.3), if βN − α < 0 or βN/α < 1, then all solutions with non-negative initial values approach the limit zero as t → ∞, (the constant solution I = N − β/α is negative), while if βN/α > 1, then all solutions with non-negative initial values except the constant solution I = 0 approach the limit N − α/β > 0 as t → ∞. Thus there is always a single limiting value for I but the value of the quantity βN/α determines which limiting value is approached, regardless of the initial state of the disease. In epidemiological terms this says that if the quantity βN/α is less than one the infection dies out in the sense that the number of infectives approaches zero. For this reason the constant solution I = 0, which corresponds to S = N, is called the disease-free equilibrium. On the other hand, if the quantity βN/α exceeds one,

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the infection persists. The constant solution I = N − α/β, which corresponds to S = α/β, is called an endemic equilibrium. The value of 1 for the quantity βN/α is a tipping point in the sense that the behavior of the solution changes if this quantity passes through 1 because of some change in the parameters of the model. The quantity βN/α also has an epidemiological interpretation. Since βN is the number of contacts made by an average infective per unit time and 1/α is the mean infective period, the quantity βN/α is the number of secondary infections caused by introducing a single infective into a wholly susceptible population. The basic reproduction number is defined as the number of secondary infections caused by an average infective introduced into a wholly susceptible population over the course of the disease. The basic reproduction number is usually denoted by R0 . Here, the basic reproduction number or contact number for the disease is R0 =

βN . α

(2.5)

In studying an infectious disease model, the basic reproduction number is a central concept and its determination is invariably an essential first step. The value one for the basic reproduction number defines a threshold at which the course of the infection changes between disappearance and persistence. It is intuitively clear that if R0 < 1 the infection should die out, while if R0 > 1 the infection should establish itself. In more highly structured models than the simple one we have developed here, the calculation of the basic reproduction number may be much more complicated, but the essential concept obtains–that of the basic reproduction number as the number of secondary infections caused by an average infective over the course of the disease. Since an endemic equilibrium corresponds to a long-term situation, it would be more realistic to include demographic processes, that is, births and natural deaths, in our model. A simple assumption is that there is a birth rate Λ(N ) depending on total population size, and a proportional death rate μ. Just as for disease recovery rates the assumption of a proportional death rate is equivalent to an assumption of an exponentially distributed life span with mean 1/μ. In the absence of disease the total population size N satisfies the differential equation N ′ = Λ(N ) − μN. At this point, it is necessary to introduce some basic definitions and results to describe the qualitative behavior of solutions of this differential equation, since it is not possible to solve the differential equation analytically. The carrying capacity of the population is the limiting population size K, satisfying Λ(K) = μK,

Λ′ (K) < μ.

2.2 The SI S Model

29

The condition Λ′ (K) < μ assures the asymptotic stability of the equilibrium population size K. It is reasonable to assume that K is the only positive equilibrium, so that Λ(N ) > μN for 0 ≤ N ≤ K. For most population models, Λ(0) = 0,

Λ′′ (N ) ≤ 0.

However, if Λ(N ) represents recruitment into a behavioral class, as would be natural for models of sexually transmitted diseases, it would be plausible to have Λ(0) > 0, or even to consider Λ(N ) to be a constant function. If Λ(0) = 0, we require Λ′ (0) > μ because if this requirement is not satisfied there is no positive equilibrium and the population would die out even in the absence of disease. A model for a disease from which infectives recover with no immunity against reinfection and that includes births and deaths is S ′ = Λ(N ) − βSI − μS + αI I ′ = βSI − αI − μI,

(2.6)

describing a population with a density-dependent birth rate Λ(N ) per unit time, a mass action contact rate, a proportional death rate μ in each class, and a recovery rate α. If we add the two equations of (2.6) and use N = S + I , we obtain N ′ = Λ(N ) − μN. Thus N approaches K as t → ∞. It is easy to verify that R0 =

βK μ+α

because a single infective introduced into a wholly susceptible population of size K causes βK new infections in unit time and the mean infective period corrected for natural mortality is 1/(μ + α). It can be shown that the endemic equilibrium of (2.6), which exists if R0 > 1, is always asymptotically stable. If R0 < 1, the system has only the diseasefree equilibrium and this equilibrium is asymptotically stable. The qualitative behavior of the model (2.6) is the same as the behavior of the model (2.2) without demographics. We were able to reduce the systems of two differential equations (2.2) to a single equation because of the assumption that the total population S + I is constant or has a constant limit as t → ∞. If there are deaths due to the disease this assumption is violated, and it would be necessary to use a two-dimensional system as a model.

30

2 Simple Compartmental Models for Disease Transmission

Let us assume that infectives recover from infection at a rate αI , while infectives die of disease at a rate dI . Then the model (2.6) is replaced by the model S ′ = Λ(N ) − βSI − μS + αI I ′ = βSI − (α + d)I − μI.

(2.7)

Now the equation for total population size is N ′ = Λ(N ) − μN − dI. The model (2.7) involves the three variables S, I, N, and we may reduce it to a twodimensional system with variables I and N by replacing S by (N − I ) to give the model I ′ = βI (N − I ) − (α + d + μ)I N ′ = Λ(N ) − μN − dI. The analysis of this model is more difficult, and rather than going through this analysis now we will shift our attention to SI R models for which the general approach to models involving systems of ordinary differential equations is more readily illustrated.

2.3 The SI R Model with Births and Deaths The SI R model of Kermack and McKendrick [18] includes births in the susceptible class proportional to total population size and a death rate in each class proportional to the number of members in the class. This model allows the total population size to grow exponentially or die out exponentially if the birth and death rates are unequal. It is applicable to such questions as whether a disease will control the size of a population that would otherwise grow exponentially. We shall return to this topic, which is important in the study of many diseases in less developed countries with high birth rates. To formulate a model in which total population size remains bounded we could follow the approach suggested by Hethcote [12] in which the total population size N is held constant by making birth and death rates equal. Such a model is S ′ = −βSI + μ(N − S) I ′ = βSI − αI − μI R ′ = αI − μR.

2.3 The SI R Model with Births and Deaths

31

Because S +I +R = N and N ′ = 0, N is constant and we can view R as determined when S and I are known. Thus we may consider the two-dimensional system S ′ = −βSI + μ(N − S)

(2.8)

′

I = βSI − αI − μI. We shall examine a more general SI R model with births and deaths for a disease that may be fatal to some infectives. For such a disease, the class R of removed members should contain only recovered members, not members removed by death from the disease. It is not possible to assume that the total population size remains constant if there are deaths due to disease; a plausible model for a disease that may be fatal to some infectives must allow the total population to vary in time. With a mass action contact rate and a density-dependent birth rate we would have a model S ′ = Λ(N ) − βSI − μS I ′ = βSI − μI − dI − αI N ′ = Λ(N ) − dI − μN.

(2.9)

If d = 0, so that there are no disease deaths, the equation for N is N ′ = Λ(N ) − μN, so that N(t) approaches a limiting population size K provided Λ′ (K) < μ so that the equilibrium K of the equation for N is asymptotically stable. We shall analyze the model (2.9) with d = 0 qualitatively. This qualitative analysis depends on the ideas of equilibria and linearization of a system about an equilibrium, a general approach dating back to the early part of the twentieth century. In view of the remark above, our analysis will also apply to the more general model (2.9) if there are no disease deaths. Analysis of the system (2.9) with d > 0 is much more difficult. We will confine our study of (2.9) to a description without details. The first stage of the analysis is to note that the model (2.9) is a properly posed problem. A properly posed problem is a problem that has a unique solution (so that solving the mathematical problem yields an expression that can have epidemiological meaning), and that the solution remains non-negative (so that it has epidemiological meaning). That is, since S ′ ≥ 0 if S = 0 and I ′ ≥ 0 if I = 0, we have S ≥ 0, I ≥ 0 for t ≥ 0 and since N ′ ≤ 0 if N = K we have N ≤ K for t ≥ 0. Thus the solution always remains in the biologically realistic region S ≥ 0, I ≥ 0, 0 ≤ N ≤ K if it starts in this region. By rights, we should verify such conditions whenever we analyze a mathematical model, but in practice this step is frequently overlooked. Our approach will be to identify equilibria (constant solutions) and then to determine the asymptotic stability of each equilibrium. As we have defined at the

32

2 Simple Compartmental Models for Disease Transmission

end of Sect. 2.1, asymptotic stability of an equilibrium means that a solution starting sufficiently close to the equilibrium remains close to the equilibrium and approaches the equilibrium as t → ∞, while instability of the equilibrium means that there are solutions starting arbitrarily close to the equilibrium that do not approach it. Equilibrium analysis of a system of two differential equations requires the idea of linearization of a system of differential equations and some matrix algebra. To find equilibria (S∞ , I∞ ) we set the right side of each of the two equations equal to zero. The second of the resulting algebraic equations factors gives two alternatives. The first alternative is I∞ = 0, which will give a disease-free equilibrium, and the second alternative is βS∞ = μ + α, which will give an endemic equilibrium, provided βS∞ = μ + α < βK. If I∞ = 0 the other equation gives S∞ = K = Λ/μ. For the endemic equilibrium the first equation gives I∞ =

μ Λ − . μ+α β

We linearize about an equilibrium (S∞ , I∞ ) by letting y = S − S∞ , z = I − I∞ , writing the system in terms of the new variables y and z and retaining only the linear terms in a Taylor expansion. We obtain a system of two linear differential equations, y ′ = −(βI∞ + μ)y − βS∞ z

z′ = βI∞ y + (βS∞ − μ − α)z.

The coefficient matrix of this linear system is   −βI∞ − μ −βS∞ . βI∞ βS∞ − μ − α This matrix is also known as the Jacobian matrix. We then look for solutions whose components are constant multiples of eλt ; this means that λ must be an eigenvalue of the coefficient matrix. The condition that all solutions of the linearization at an equilibrium tend to zero as t → ∞ is that the real part of every eigenvalue of this coefficient matrix is negative. At the disease-free equilibrium the matrix is   −μ −βK , 0 βK − μ − α which has eigenvalues −μ and βK − μ − α. Thus, the disease-free equilibrium is asymptotically stable if βK < μ + α and unstable if βK > μ + α. Note that this condition for instability of the disease-free equilibrium is the same as the condition for the existence of an endemic equilibrium. In general, the condition that the eigenvalues of a 2 × 2 matrix have negative real part is that the determinant be positive and the trace (the sum of the diagonal

2.3 The SI R Model with Births and Deaths

33

elements) be negative. Since βS∞ = μ + α at an endemic equilibrium, the Jacobian matrix, or matrix of the linearization at an endemic equilibrium is   −βI∞ − μ −βS∞ βI∞ 0 and this matrix has positive determinant and negative trace. Thus, the endemic equilibrium, if there is one, is always asymptotically stable. If the quantity R0 =

K βK = μ+α S∞

is less than one, then the system has only the disease-free equilibrium and this equilibrium is asymptotically stable. In fact, it is not difficult to prove that this asymptotic stability is global, that is, that every solution approaches the disease-free equilibrium. If the quantity R0 is greater than one, then the disease-free equilibrium is unstable, but there is an endemic equilibrium that is asymptotically stable. Again, the quantity R0 is the basic reproduction number. It depends on the particular disease (determining the parameter α) and on the rate of contacts, which may depend on the population density in the community being studied. The disease model exhibits a threshold behavior: If the basic reproduction number is less than one, then the disease will die out, but if the basic reproduction number is greater than one, then the disease will be endemic. Just as for the SI S model of the preceding section, the basic reproduction number is the number of secondary infections caused by a single infective introduced into a wholly susceptible population because the number of contacts per infective in unit time is βK, and the mean infective period (corrected for natural mortality) is 1/(μ + α). There are two aspects of the analysis of the model (2.9) that are more complicated than the analysis of (2.8). The first is in the study of equilibria. Because of the dependence of Λ(N ) on N , it is necessary to use two of the equilibrium conditions to solve for S and I in terms of N and then substitute this into the third condition to obtain an equation for N. Then by comparing the two sides of this equation for N = 0 and N = K it is possible to show that there must be an endemic equilibrium value of N between 0 and K if R0 > 1. The second complication is in the stability analysis. Since (2.9) is a threedimensional system that cannot be reduced to a two-dimensional system, the coefficient matrix of its linearization at an equilibrium is a 3 × 3 matrix and the resulting characteristic equation is a cubic polynomial equation of the form λ3 + a1 λ2 + a2 λ + a3 = 0 . The Routh–Hurwitz conditions [16, 22] a1 > 0,

a1 a2 > a3 > 0

34

2 Simple Compartmental Models for Disease Transmission

are necessary and sufficient conditions for all roots of the characteristic equation to have negative real part. A technically complicated calculation is needed to verify that these conditions are satisfied at an endemic equilibrium for the model (2.9). The asymptotic stability of the endemic equilibrium means that the compartment sizes approach a steady state. If the equilibrium had been unstable, there would have been a possibility of sustained oscillations. Oscillations in a disease model mean fluctuations in the number of cases to be expected. If the oscillations have a long period, then this could mean that experimental data for a short period would be quite unreliable as a predictor of the future. Epidemiological models that incorporate additional factors may exhibit oscillations. A variety of such situations is described in [14, 15]. From the third equation of (2.9) we obtain N ′ = Λ − μN − αI, where N = S + I + R. From this, we see that at the endemic equilibrium N = Λ/μ − αI /μ, and the reduction in the population size from the carrying capacity K is   d α αK . I∞ = − dμ μ+α β The parameter α in the SI R model may be considered as describing the pathogenicity of the disease. If α is large, it is less likely that R0 > 1. If α is small, then the total population size at the endemic equilibrium is close to the carrying capacity K of the population. Thus, the maximum population decrease caused by disease will be for diseases of intermediate pathogenicity with α not close to either 0 or 1. Numerical simulations indicate that the approach to endemic equilibrium for an SI R model is like a rapid and severe epidemic if the epidemiological and demographic time scales are very different. The same happens in the SI S model. If there are few disease deaths, the number of infectives at endemic equilibrium may be substantial, and there may be damped oscillations of large amplitude about the endemic equilibrium. For both the SI R and SI S models we may write the differential equation for I as I ′ = I [β(N)S − (μ + α)] = β(N)I [S − S∞ ], which implies that whenever S exceeds its endemic equilibrium value S∞ , I is increasing and epidemic-like behavior is possible. If R0 < 1 and S < K, it follows that I ′ < 0, and thus I is decreasing. Thus, if R0 < 1, I cannot increase and no epidemic can occur.

2.4 The Simple Kermack–McKendrick Epidemic Model

35

2.4 The Simple Kermack–McKendrick Epidemic Model One of the early triumphs of mathematical epidemiology was the formulation of a simple model by Kermack and McKendrick in 1927 [17] whose predictions are very similar to the behavior, observed in countless epidemics, of disease that invade a population suddenly, grow in intensity, and then disappear leaving part of the population untouched. The Kermack–McKendrick model is a compartmental model based on relatively simple assumptions on the rates of flow between different classes of members of the population. The SARS epidemic of 2002–2003 revived interest in epidemic models, which had been largely ignored since the time of Kermack and McKendrick, in favor of models for endemic diseases. The special case of the model proposed by Kermack and McKendrick in 1927 which is the starting point for our study of epidemic models is S ′ = −βSI I ′ = βSI − αI R ′ = αI.

(2.10)

A flow chart is shown in Fig. 2.1. It is based on the same assumptions that were made for the SI S model of Sect. 2.2, except that recovered infectives go to a removed class rather than returning to the susceptible class. We will refer to this model as the simple Kermack– McKendrick epidemic model for convenience, but we remind the reader that it is a very special case of the actual Kermack–McKendrick epidemic model. From (2.10) we see that N = S + I + R is constant. The assumptions of a constant rate of contacts and of an exponentially distributed recovery rate are unrealistically simple. More general models can be constructed and analyzed, but our goal here is to show what may be deduced from extremely simple models. It will turn out that many more realistic models exhibit very similar qualitative behaviors.

Fig. 2.1 Flow chart for the SI R model (2.10)

36

2 Simple Compartmental Models for Disease Transmission

In our model, R is determined once S and I are known, and we can drop the R equation from our model, leaving the system of two equations S ′ = −βSI I ′ = βSI − αI,

(2.11)

together with initial conditions S(0) = S0 ,

I (0) = I0 ,

S0 + I0 = N.

We think of introducing a small number of infectious individuals into a population of susceptibles and ask whether there will be an epidemic. We remark that the model makes sense only so long as S(t) and I (t) remain non-negative. Thus if either S(t) or I (t) reaches zero we consider the system to have terminated. We observe that S ′ < 0 for all t and I ′ > 0 if and only if βS/α > 1. Thus I increases so long as βS/α > 1 but since S decreases for all t, I ultimately decreases and approaches zero. The quantity R0 = βN/α determines whether there is an epidemic. If R0 < 1, the infection dies out because I ′ (t) < 0 for all t, and there is no epidemic. Ordinarily, S0 ≈ N. If the epidemic is started by a member of the population being studied, for example by returning from travel with an infection acquired away from home, we would have I0 > 0, S0 +I0 = N. A second way would be for an epidemic to be started by a visitor from outside the population. In this case, we would have S0 = N. If R0 > 1, I increases initially and this is interpreted as saying that there is an epidemic. Since (2.11) is a two-dimensional autonomous system of differential equations, the natural approach would be to find equilibria and linearize about each equilibrium to determine its stability. However, since every point with I = 0 is an equilibrium, the system (2.11) has a line of equilibria and this approach is not applicable (the linearization matrix at each equilibrium has a zero eigenvalue). The standard linearization theory for systems of ordinary differential equations is not applicable, and it is necessary to develop a new mathematical approach. The sum of the two equations of (2.11) is (S + I )′ = −αI. Thus S + I is a non-negative smooth decreasing function and therefore tends to a limit as t → ∞. Also, it is not difficult to prove that the derivative of a smooth decreasing function which is bounded below must tend to zero, and this shows that I∞ = lim I (t) = 0. t→∞

Thus S + I has limit S∞ .

2.4 The Simple Kermack–McKendrick Epidemic Model

37

Integration of the sum of the two equations of (2.11) from 0 to ∞ gives −



∞ 0

(S(t) + I (t))′ dt = S0 + I0 − S∞ = N − S∞ = α



∞

I (t)dt.

0

Division of the first equation of (2.11) by S and integration from 0 to ∞ gives S0 log =β S∞



∞

I (t)dt

0

β [N − S∞ ] α   S∞ . = R0 1 − N

=

(2.12)

Equation (2.12) is called the final size relation. It gives a relation between the basic reproduction number and the size of the epidemic. Note that the final size of the epidemic, the number of members of the population who are infected over the course of the epidemic, is N − S∞ . This is often described in terms of the attack rate (1 − S∞ /N). [Technically, the attack rate should be called an attack ratio, since it is dimensionless and is not a rate]. The attack rate is the fraction of the population that becomes infected over the course of the epidemic. The final size relation (2.12) can be generalized to epidemic models with more complicated compartmental structure than the simple SI R model (2.11), including models with exposed periods, treatment models, and models including quarantine of suspected individuals and isolation of diagnosed infectives. The original Kermack–McKendrick model [17] included dependence on the time since becoming infected (age of infection), and this includes such models. We will describe this generalization in Chap. 4. Integration of the first equation of (2.11) from 0 to t gives log

S0 =β S(t) =



t

I (t)dt

0

β [N − S(t) − I (t)], α

and this leads to the form I (t) + S(t) −

α α log S(t) = N − log S0 . β β

(2.13)

This implicit relation between S and I describes the orbits of solutions of (2.11) in the (S, I ) plane. In addition, since the right side of (2.12) is finite, the left side is also finite, and this shows that S∞ > 0. It is not difficult to prove that there is a unique solution of the final size relation (2.12). To see this, we define the function

38

2 Simple Compartmental Models for Disease Transmission

g(x) = log

 x S0 − R0 1 − . x N

Then g(0+) > 0,

g(N) < 0,

and g ′ (x) < 0 if and only if 0 1, g(x) is monotone decreasing from a positive value at x = 0+ to a minimum at x = N/R0 and then increases to a negative value at x = N0 . Thus there is a unique zero S∞ of g(x) with S∞
0, we actually have g

Fig. 2.2 Graph of the function g(x)



S0 R0



< 0,

2.4 The Simple Kermack–McKendrick Epidemic Model

39

and S∞
1−

1 α =1− . βN R0

Initially, the number of infectives grows exponentially because the equation for I may be approximated by I ′ = (βN − α)I and the initial growth rate is r = βN − α = α(R0 − 1) .

40

2 Simple Compartmental Models for Disease Transmission

This initial growth rate r may be estimated from incidence data when an epidemic begins. Since N and α may be measured, β may be calculated as β=

r +α . N

However, because of incomplete data and under-reporting of cases, this estimate may not be very accurate. This inaccuracy is even more pronounced for an outbreak of a previously unknown disease, where early cases are likely to be mis-diagnosed. Because of the final size relation, estimation of β or R0 is an important question that has been studied by a variety of approaches. Estimation of the initial growth rate from data can provide an estimate of the contact rate β. However, this relation is valid only for the model (2.11) and does not hold for models with different compartmental structure, such as an exposed period. If βS0 > α, I increases initially to a maximum number of infectives when the derivative of I is zero, that is, when S = α/β. This maximum is given by Imax = S0 + I0 −

α α α α log S0 − + log , β β β β

(2.15)

obtained by substituting S = α/β, I = Imax into (2.13). Example 1 A study of Yale University freshmen [10] reported by Hethcote [13] described an influenza epidemic with S0 = 0.911, S∞ = 0.513. Here we are measuring the number of susceptibles as a fraction of the total population size, or using the population size K as the unit of size. Substitution into the final size relation gives the estimate βN/α = 1.18 and R0 = 1.18. Since we know that 1/α is approximately 3 days for influenza, we see that βN is approximately 0.39 contacts per day per member of the population. Example 2 (The Great Plague in Eyam) The village of Eyam near Sheffield, England suffered an outbreak of bubonic plague in 1665–1666 the source of which is generally believed to be the Great Plague of London. The Eyam plague was survived by only 83 of an initial population of 350 persons. As detailed records were preserved and as the community was persuaded to quarantine itself to try to prevent the spread of disease to other communities, the disease in Eyam has been used as a case study for modeling [21]. Detailed examination of the data indicates that there were actually two outbreaks of which the first was relatively mild. Thus we shall try to fit the model (2.11) over the period from mid-May to mid-October 1666, measuring time in months with an initial population of 7 infectives and 254 susceptibles, and a final population of 83. Raggett [21] gives values of susceptibles and infectives in Eyam on various dates, beginning with S(0) = 254, I (0) = 7, shown in Table 2.1. The final size relation with S0 = 254, I0 = 7, S∞ = 83 gives β/α = 6.54×10−3 , α/β = 153. The infective period was 11 days, or 0.3667 month, so that α = 2.73. Then β = 0.0178. The relation (2.15) gives an estimate of 30.4 for the maximum

2.4 The Simple Kermack–McKendrick Epidemic Model Table 2.1 Eyam Plague data

Date (1666) July 3/4 July 19 August 3/4 August 19 September 3/4 September 19 October 4/5 October 20

41 Susceptibles 235 201 153.5 121 108 97 Unknown 83

Infectives 14.5 22 29 21 8 8 Unknown 0

Fig. 2.3 The S–I plane

number of infectives. We use the values obtained here for the parameters β and α in the model (2.11) for simulations of the phase plane, the (S, I ) plane, and for graphs of S and I as functions of t (Figs. 2.3 and 2.4). Figure 2.5 plots these data points together with the phase portrait given in Fig. 2.3 for the model (2.11). The actual data for the Eyam epidemic are remarkably close to the predictions of this very simple model. However, the model is really too good to be true. Our model assumes that infection is transmitted directly between people. While this is possible, bubonic plague is transmitted mainly by rat fleas. When an infected rat is bitten by a flea, the flea becomes extremely hungry and bites the host rat repeatedly, spreading the infection in the rat. When the host rat dies its fleas move on to other rats, spreading the disease further. As the number of available rats decreases the fleas move to human hosts, and this is how plague starts in a human population (although the second phase of the epidemic may have been the pneumonic form of bubonic plague, which can be spread from person to person). One of the main reasons for the spread of plague from Asia into Europe was the passage of many trading ships; in medieval times ships were invariably infested with rats. An accurate model of plague transmission would have to include flea and rat populations, as well as movement in space. Such a model would be extremely complicated and its predictions might well not be any closer to observations than our simple unrealistic model. Raggett

42

2 Simple Compartmental Models for Disease Transmission

Fig. 2.4 Time plot of S and I

Fig. 2.5 The S–I plane, model and data

also used a stochastic model to fit the data, but the fit was rather poorer than the fit for the simple deterministic model (2.11). In the village of Eyam the rector persuaded the entire community to quarantine itself to prevent the spread of disease to other communities. One effect of this policy was to increase the infection rate in the village by keeping fleas, rats, and people in close contact with one another, and the mortality rate from bubonic plague was much higher in Eyam than in London. Further, the quarantine could do nothing to prevent the travel of rats and thus did little to prevent the spread of disease to other communities. One message this suggests to mathematical modelers is that control strategies based on false models that do not describe the epidemiological

2.5 Epidemic Models with Deaths due to Disease

43

situation accurately may be harmful, and it is essential to distinguish between assumptions that simplify but do not alter the predicted effects substantially, and wrong assumptions which make an important difference.

2.5 Epidemic Models with Deaths due to Disease So far, we have analyzed only models with no disease deaths, for which it may be assumed that the total population size is constant. The assumption in the model (2.11) of a rate of contacts per infective which is proportional to population size N , called mass action incidence or bilinear incidence, was used in all the early epidemic models. However, it is quite unrealistic, except possibly in the early stages of an epidemic in a population of moderate size. It is more realistic to assume a contact rate which is a non-increasing function of the total population size. For example, a situation in which the number of contacts per infective in unit time is constant, called standard incidence, is a more accurate description for sexually transmitted diseases. If there are no disease deaths, so that the total population size remains constant, such a distinction is unnecessary. We generalize the model (2.11) by assuming that an average member of the population makes Nβ(N) contacts in unit time [4, 8]. It is reasonable to assume β ′ (N ) ≤ 0 to express the idea of saturation in the number of contacts. Then mass action incidence corresponds to the choice a(N) = βN , and standard incidence corresponds to the choice a(N) = a. The assumptions a(N) = Nβ(N ), a ′ (N ) ≥ 0 imply that β(N) + Nβ ′ (N ) ≥ 0. Some epidemic models [8] have used a Michaelis–Menten type of interaction of the form β(N) =

a . 1 + bN

Another form based on a mechanistic derivation for pair formation [11] leads to an expression of the form β(N) =

1 + bN +

a . √ 1 + 2bN

Data for diseases transmitted by contact in cities of moderate size [20] suggests that data fits the assumption of a form β(N) = aN p−1

44

2 Simple Compartmental Models for Disease Transmission

with p = 0.05 quite well. All of these forms satisfy the conditions β ′ (N ) ≤ 0 and (2.5). Because the total population size is now present in the model, we must include an equation for total population size in the model. This forces us to make a distinction between members of the population who die of the disease and members of the population who recover with immunity against reinfection. We assume, as in the previous chapter, a recovery rate αI and a disease death rate dI . We use S, I , and N as variables, with N = S + I + R. We now obtain a three-dimensional model S ′ = −β(N)SI I ′ = β(N)SI − (α + d)I N ′ = −dI.

(2.16)

Since N is now a decreasing function, we define N (0) = N0 = S0 + I0 . We also have the equation R ′ = αI , but we do not need to include it in the model since R is determined when S, I, and N are known. We should note that if d = 0 the total population size remains equal to the constant N, and the model (2.16) reduces to the simpler model (2.11) with β(N) replaced by the constant β(N0 ). We wish to show that the model (2.16) has the same qualitative behavior as the model (2.11), namely that there is a basic reproduction number which distinguishes between disappearance of the disease and an epidemic outbreak, and that some members of the population are left untouched when the epidemic passes. These two properties are the central features of all epidemic models. For the model (2.16) the basic reproduction number is given by R0 =

N0 β(N0 ) α+d

because a single infective introduced into a wholly susceptible population makes c(N0 ) = N0 β(N0 ) contacts in unit time, all of which are with susceptibles and thus produce new infections, and the mean infective period, corrected for mortality, is 1/(α + d). We assume that β(0) is finite, thus ruling out standard incidence (standard incidence does not appear to be realistic if the total population N approaches zero, and it would be more natural to assume that c(N) grows linearly with N for small N ). If we let t → ∞ in the sum of the first two equations of (2.16) we obtain (α + d)



∞ 0

I (s)ds = S0 + I0 − S∞ = N − S∞ .

The first equation of (2.16) may be written as −

S ′ (t) = β(N (t))I (t). S(t)

2.5 Epidemic Models with Deaths due to Disease

45

Since β(N) ≥ β(N0 ), integration from 0 to ∞ gives



S0 = S∞

log

∞

β(N(t))I (t)dt

0

≥ β(N0 )



∞

I (t)dt

0

β(N0 )(N0 − S∞ ) . (α + d)N0

= We now obtain a final size inequality S0 = log S∞



∞

β(N(t))I (t)dt

0

≥ β(N0 )



∞ 0

  S∞ . I (t)dt = R0 1 − N0

If the case fatality ratio d/(d +α) is small, the final size inequality is an approximate equality. It is not difficult to show that N (t) ≥ αN0 /(α + d) and then a similar calculation using the inequality β(N) ≤ β(αN0 /(α + d)) < ∞ shows that S0 log ≤ β(αN0 /(α + d)) S∞



∞

I (t)dt,

0

from which we may deduce that S∞ > 0. It is important to be able to estimate the fraction of infectives who die of disease over the course of the epidemic. At every time, the rate of recovery is αI and the rate of disease deaths is dI , and thus the mortality fraction at each time is d . d +α This is what is sometimes described (incorrectly because it is not a rate) as the epidemic death rate. While the mortality fraction overall is d/(d +α), reports during the epidemic would underestimate the mortality fraction because there may be some infectives who would die later during the epidemic. On the other hand, for a disease like influenza, where many cases are mild enough to go unreported, reports would tend to overestimate the mortality fraction.

46

2 Simple Compartmental Models for Disease Transmission

2.6 *Project: Discrete Epidemic Models The discrete analogue of the continuous-time epidemic model (2.11) is Sj +1 = Sj Gj ,

 Ij +1 = Sj 1 − Gj + σ Ij , Gj = e

−βIj /N

,

(2.17)

j = 1, 2, . . . ,

where Sj and Ij denote the numbers of susceptible and infective individuals at time j , respectively. Here, Gj is the probability that a susceptible individual at time j will remain susceptible to time j + 1, that is, will not be infected between timej and time j + 1, and σ = e−α is the probability that an infected individual at time j will remain infected to time j + 1. Assume that the initial conditions are S(0) = S0 > 0, I (0) = I0 > 0, and S0 + I0 = N. Exercise 1 Consider the system (2.17). (a) Show that the sequence {Sj + Ij } has a limit S∞ + I∞ = lim (Sj + Ij ). j →∞

(b) Show that I∞ = lim Ij = 0. j →∞

(c) Show that log

∞ Im S0 . =β S∞ N m=0

(d) Show that   S0 S∞ log , = R0 1 − S∞ N β with R0 = 1−σ . Next, consider the case that there are k infected stages and there is treatment in some stages, with treatment rates that can be different in different stages. Assume that selection of members for treatment occurs only at the beginning of (i) (i) a stage. Let Ij and Tj denote the numbers of infected and treated individuals, respectively, in stage i (i = 1, 2, . . . , k) at time j . Let σiI denote the probability

2.7 *Project: Pulse Vaccination

47

that an infected individual in the I (i) stage continues on to the next stage, either treated or untreated, and let σiT denote the probability that an individual in the T (i) stage continues on to the next treated stage. In addition, of the members leaving an infected stage I (i) , a fraction pi enters treatment in T (i+1) , while the remaining fraction qi continues to I (i+1) . Let mi denote the fraction of infected members who go through the stage I (i) , and ni the fraction of infected members who go through the stage T (i) . Then, m1 = q1 , m2 = q1 q2 , . . . , mk = q1 q2 · · · qk , n1 = p1 , n2 = p1 + q1 p2 , . . . , nk = p1 + q1 p2 + . . . + q1 q2 · · · qk−1 pk . The discrete system with treatment is Sj +1 = Sj Gj , I (1) Ij(1) +1 = q1 Sj (1 − Gj ) + σ1 Ij , (1)

(1)

Tj +1 = p1 Sj (1 − Gj ) + σ1T Tj (i)

(i−1)

(2.18) (i)

I Ij +1 = qi (1 − σi−1 )Ij

+ σiI ηi Ij ,

(i−1) I T Tj(i) + (1 − σi−1 )Tj(i−1) + σiT Tj(i) , +1 = pi (1 − σi−1 )Ij

[i = 2, . . . , k, j ≥ 0], with Gj = e−β



k

i=1

(i)

(i)

ǫi Ij /N +δi Tj /N



,

where ǫi is the relative infectivity of untreated individuals at stage i and δi is the relative infectivity of treated individuals at stage i. (e) Show that Rc = βN

k i=1



δi ni ǫi m i + I 1 − σi 1 − σiT



and that   S0 S∞ log . = Rc 1 − S∞ N

2.7 *Project: Pulse Vaccination Consider an SIR model (2.9) with Λ = μK. For measles, typical parameter choices are μ = 0.02, β = 1800, α = 100, K = 1 (to normalize carrying capacity to 1) [9].

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2 Simple Compartmental Models for Disease Transmission

Question 1 Show that for these parameter choices R0 ≈ 18 and to achieve herd immunity would require vaccination of about 95% of the susceptible population. In practice, it is not possible to vaccinate 95% of a population because not all members of the population would come to be vaccinated and not all vaccinations are successful. One way to avoid recurring outbreaks of disease is “pulse vaccination” [1, 24, 25]. The basic idea behind pulse vaccination is to vaccinate a given fraction p of the susceptible population at intervals of time T with T (depending on p) chosen to ensure that the number of infectives remains small and approaches zero. In this project we will give two approaches to the calculation of a suitable function T (p). The first approach depends on the observation that I decreases so long as S < Γ < (μ + α)/β. We begin by vaccinating pΓ members, beginning with S(0) = (1 − p)Γ . From (8.7), S ′ = μK − μS − βSI ≥ μK − μS. Then S(t) is greater than the solution of the initial value problem S ′ = μK − μS,

S(0) = (1 − p)Γ.

Question 2 Solve this initial value problem and show that the solution obeys S(t) < Γ, 0 ≤ t
0, I2∗ > 0) equilibria? (j) Simulate the full system (2.19) when the demographic equation is in the period doubling regime, where there are orbits with periods of double the length of periods for smaller parameter values. What are your conclusions? References: [5, 6].

2.9 Project: An Epidemic Model in Two Patches Consider the following SIS model with dispersion between two patches, Patch 1 and Patch 2, where in Patch i ∈ {1, 2} at generation t, Si (t) denotes the population of susceptible individuals; Ii (t) denotes the population of infected assumed infectious; Ti (t) ≡ Si (t) + Ii (t) denotes the total population size. The constant dispersion coefficients DS and DI measure the probability of dispersion by the susceptible and infective individuals, respectively. Observe that we are using a different notation from what we have used elsewhere, writing variables as a function of t rather than using a subscript for the independent variable in order to avoid needing double subscripts: S1 (t + 1) = I1 (t + 1) = S2 (t + 1) = I2 (t + 1) =

(1 − DS ) S1 (t) + DS  S2 (t), (1 − DI )I1 (t) + DI I2 (t), DS  S1 (t) + (1 − DS ) S2 (t), DI I1 (t) + (1 − DI )I2 (t),

2.10 Project: Fitting Data for an Influenza Model

53

where

and

  −αi Ii (t)  + γi Ii (t)(1 − σi ), Si (t) = fi (Ti (t)) + γi Si (t) exp Ti (t)   −αi Ii (t) Si (t) + γi σi Ii (t) Ii (t) = γi (1 − exp Ti (t) 0 ≤ γi , σi , αi , DS , DI ≤ 1.

Let fi (Ti (t)) = Ti (t) exp(ri − Ti (t)), where ri is a positive constant. (a) Using computer explorations, determine whether it is possible to have a globally stable disease-free equilibrium on a patch (without dispersal) where the full system with dispersal has a stable endemic equilibrium. Do you have a conjecture? (b) Using computer explorations determine whether it is possible to have a globally stable endemic equilibrium on a patch (without dispersal) where the full system with dispersal has a stable disease-free equilibrium. Do you have a conjecture? References:[3, 7, 23].

2.10 Project: Fitting Data for an Influenza Model Consider an SIR model (2.11) with basic reproduction number 1.5. 1. Describe the qualitative changes in (S, I, R) as a function of time for different values of β and α with β ∈ {0.0001, 0.0002, . . . , 0.0009}, for the initial condition (S, I, R) = (106 , 1, 0). 2. Discuss the result of part (a) in terms of the basic reproduction number (what is β/γ ?). Use a specific disease such as influenza to provide simple interpretations for the different time courses of the disease for the different choices of β and γ . 3. Repeat the steps in part (a) for values of R0 ∈ {1.75, 2, 2.5}, and for each value of R0 , choose the best pair of values (β, α) that fits the slope before the first peak in the data found in Table 2.2 for reported H1N1 influenza cases in México. (Hint: normalize the data so that the peak is 1, and then multiply the data by the size of the peak in the simulations.)

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2 Simple Compartmental Models for Disease Transmission

Table 2.2 Reported cases for H1N1-pandemic in Mexico Day 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94

Cases 2 1 3 2 3 3 4 4 5 7 3 1 2 5 7 4 10 11 13 4

Day 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114

Cases 4 11 5 7 4 4 4 11 17 26 20 12 26 33 44 107 114 155 227 280

Day 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134

Cases 318 399 412 305 282 227 212 187 212 237 231 237 176 167 139 142 162 138 117 100

Day 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154

Cases 83 75 87 98 71 73 78 67 68 69 65 85 55 67 75 71 97 168 126 148

Day 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174

Cases 152 138 159 186 222 204 257 208 198 193 243 231 225 239 219 199 215 309 346 332

Day 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194

Cases 328 298 335 330 375 366 291 251 215 242 223 317 305 228 251 207 159 155 214 237

2.11 Project: Social Interactions Suppose we have a system with multiple classes of mathematical biology teachers (MBT) at time t. The classes roughly capture the MBT individual attitudes toward learning new stuff. “Reluctant” means the class of MBTs that come into the door as new hires without a disposition to learn new stuff; the positive class corresponds to those who join the MBTs with the right attitude; and the rest of the classes should be self-explanatory. The total population is divided into five classes of MBT individuals: positive (P ), reluctant (R), masterful (M), unchangeable (that is, negative) (U ), and inactive (I ). Assume that N (t) = R(t) + P (t) + M(t) + U (t) + I (t) and that the total number of MBTs is constant, that is, N (t) = K μ for all t, where K is a constant. The model is P ′ = qK − βP

M + δR − μP K

R ′ = (1 − q)K − (δ + μ)R − αR M ′ = βP

M − (γ + μ)M K

2.12 Exercises

55

U ′ = −μU + αR I ′ = γ M − μI,

where q, β, δ, μ, γ , and α are constants and 0 ≤ q ≤ 1. 1. 2. 3. 4.

Interpret the parameters. Look at the stability of the simplest equilibrium (your choice). From R0 , discuss what would be the impact of changing parameters q, γ , and δ. What are your conclusions from this model?

2.12 Exercises 1. Find the basic reproduction number and the endemic equilibrium for the SI S model S ′ = −βSI + αI

I ′ = βSI − (α + d)I,

with a death rate dI due to disease. 2. Find the basic reproduction number for the SI S model (2.7) that includes births, natural deaths, and disease deaths. Show that the disease-free equilibrium is asymptotically stable if and only if R0 < 1, and that if R0 > 1 there is an asymptotically stable endemic equilibrium. 3. Modify the SI S model (2.2) to the situation in which there are two competing strains of the same disease, generating two infective classes I1 , I2 under the assumption that coinfections are not possible. Does the model predict coexistence of the two strains or competitive exclusion? 4.∗ A communicable disease from which infectives do not recover may be modeled by the pair of differential equations S ′ = −βSI,

I ′ = βSI.

Show that in a population of fixed size K such a disease will eventually spread to the entire population. 5.∗ Consider a disease spread by carriers who transmit the disease without exhibiting symptoms themselves. Let C(t) be the number of carriers and suppose that carriers are identified and isolated from contact with others at a constant per capita rate α, so that C ′ = −αC. The rate at which susceptibles become infected is proportional to the number of carriers and to the number of susceptibles, so that S ′ = −βSC. Let C0 and S0 be the numbers of carriers and susceptibles, respectively, at time t = 0.

56

2 Simple Compartmental Models for Disease Transmission

(a) Determine the number of carriers at time t from the equation for C. (b) Substitute the solution to part (a) into the equation for S and determine the number of susceptibles at time t. (c) Find limt→∞ S(t), the number of members of the population who escape the disease. 6.∗ Consider a population of fixed size K in which a rumor is being spread by word of mouth. Let y(t) be the number of people who have heard the rumor at time t and assume that everyone who has heard the rumor passes it on to r others in unit time. Thus, from time t to time (t + h) the rumor is passed on hry(t) times, but a fraction y(t)/K of the people who hear it have already heard it, and thus  K−y(t) people who hear the rumor for the first time. there are only hry(t) K Use these assumptions to obtain an expression for y(t + h) − y(t), divide by h, and take the limit as h → 0 to obtain a differential equation satisfied by y(t). 7. At 9 AM one person in a village of 100 inhabitants starts a rumor. Suppose that everyone who hears the rumor tells one other person per hour. Using the model of the previous exercise, determine how long it will take until half the village has heard the rumor. 8.∗ If a fraction λ of the population susceptible to a disease that provides immunity against reinfection moves out of the region of an epidemic, the situation may be modeled by a system S ′ = −βSI − λS,

I ′ = βSI − αI.

Show that both S and I approach zero as t → ∞. 9. Consider the basic SI R model. We now consider a vaccination class in place of recovery: S ′ (t) = μN − βS NI − (μ + φ)S I ′ (t) = βS NI − (μ + γ )I V ′ (t) = γ I + φS − μV .

(2.22)

(a) Show that dN dt = 0. What does this result imply? (b) Discuss why it is enough to study the first two equations. (c) Let R0 (φ) be R0 when φ = 0, and R0 (0) be R0 when φ = 0. Compute R0 (φ). What is the value of R0 (0)? Compare R0 (φ) with R0 (0). (d) Compute the equilibria. (e) Do the local stability analysis of the disease-free equilibrium state and the endemic state. 10. In cases of constant recruitment, the limiting system as t → ∞ and the original system usually have the same qualitative dynamics. We thus consider our previous SIR model with vaccination (refer to problem 9) changing the recruitment rate from the constant μN to the constant Λ. This means that a

2.12 Exercises

57

certain fixed number of individuals join or arrive into the susceptible class per unit time. The model becomes S ′ (t) = Λ − βS NI − μS I ′ (t) = βS NI − (μ + γ )I V ′ (t) = γ I − μV ,

(2.23)

where, again, N (t) = S(t) + I (t) + V (t).

(a) What are the units of Λ, βS NI , μ, γ , β, and μS? (b) Find the equation for N ′ (t) where N = S + I + V , and solve this equation for N (t). Observe that the population size for this model is not constant. (c) Show that N (t) → Λ μ as t → ∞. (d) Consider the limiting system S ′ (t) = μN − βSI − μS I ′ (t) = βSI − (μ + γ )I V ′ (t) = γ I − μV .

(2.24)

Explain why it is enough to consider the first two equations as V (t) = N − S(t) − I (t) when studying the dynamics of this limiting system, find R0 , and do the local stability analysis of the equilibria. 11. An epidemic model with two latent classes is described by the following system of ODE’s: S ′ = Λ − βS NI − μS L′1 = pβS NI − (μ + k1 + r1 )L1 L′2 = (1 − p)βS NI − (μ + k2 + r2 )L2 I ′ = k1 L1 + k2 L2 − (μ + r3 )I, where N = S + L1 + L2 + I. A fraction p, 0 < p < 1, goes into the first latent class and a fraction 1 − p goes into the second latent class. The two classes have different rates of progression to the infectious stage. Compute the basic reproduction number R0 . 12. If vaccination strategies are incorporated for newborns, we assume that not every new birth is susceptible. Suppose that the per capita vaccination rate is p; a newborn is vaccinated with probability p. The modified model is S ′ (t) = (1 − p)μN − βS NI − μS I ′ (t) = βS NI − (μ + γ )I V ′ (t) = γ I − μV + pμN.

(2.25)

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2 Simple Compartmental Models for Disease Transmission

(a) Calculate the Jacobian matrix of (2.25) at the disease-free equilibrium points? (b) Find the corresponding eigenvalues of the above matrix. (c) Find the basic reproduction numbers (R0 ). (d) Study the stability of the disease-free equilibrium points of model (2.25). Although N is constant and we could reduce this to a 2-D system, derive the stability of the full 3 × 3 system by using the Routh–Hurwitz criteria. 13. The same survey of Yale students described in Example 1, Sect. 2.4 reported that 91.1% were susceptible to influenza at the beginning of the year and 51.4% were susceptible at the end of the year. Estimate the basic reproduction number and decide whether there was an epidemic. 14. What fraction of Yale students in Exercise 13 would have had to be immunized to prevent an epidemic? 15. What was the maximum number of Yale students in Exercises 13 and 14 suffering from influenza at any time? 16. An influenza epidemic was reported at an English boarding school in 1978 that spread to 512 of the 763 students. Estimate the basic reproduction number. 17. What fraction of the boarding school students in Exercise 16 would have had to be immunized to prevent an epidemic? 18. What was the maximum number of boarding school students in Exercises 16 and 17 suffering from influenza at any time? 19. A disease is introduced by two visitors into a town with 1200 inhabitants. An average infective is in contact with 0.4 inhabitants per day. The average duration of the infective period is 6 days, and recovered infectives are immune against reinfection. How many inhabitants would have to be immunized to avoid an epidemic? 20. Consider a disease with βN = 0.4, 1/α = 6 days in a population of 1200 members. Suppose the disease conferred immunity on recovered infectives. How many members would have to be immunized to avoid an epidemic? 21. A disease begins to spread in a population of 800. The infective period has an average duration of 14 days and the average infective is in contact with 0.1 persons per day. What is the basic reproduction number? To what level must the average rate of contact be reduced so that the disease will die out? 22. European fox rabies is estimated to have a transmission coefficient β of 80 km2 years/fox (assuming mass action incidence),and an average infective period of 5 days. There is a critical carrying capacity Kc measured in foxes per km2 , such that in regions with fox density less than Kc , rabies tends to die out, while in regions with fox density greater than Kc , rabies tends to persist. Use a simple Kermack–McKendrick epidemic model to estimate Kc . [Remark: It has been suggested in Great Britain that hunting to reduce the density of foxes below the critical carrying capacity would be a way to control the spread of rabies.] 23. A large English estate has a population of foxes with a density of 1.3 foxes/km2 . A large fox hunt is planned to reduce the fox population enough to prevent an outbreak of rabies. Assuming that the contact number β/α is 1 km2 /fox, find what fraction of the fox population must be caught.

2.12 Exercises

59

24. Following a complaint from the SPCA, organizers decide to replace the fox hunt of Exercise 23 by a mass inoculation of foxes for rabies. What fraction of the fox population must be inoculated to prevent a rabies outbreak? 25. What actually occurs on the estate of these exercises is that 10% of the foxes are killed and 15% are inoculated. Is there danger of a rabies outbreak? 26. Let S, I , and R represent the densities of susceptible, infected, and recovered individuals. Suppose that recovered individuals can become susceptible again after some time. The model equations are dS = −βSI + θ R dt dI = βSI − αI dt dR = αI − θ R dt where N = S + I + R and β, θ , and α are the infection, loss of immunity, and recovery rates, respectively. (a) Reduce this model to a two-dimensional system of equations. Don’t forget to show that N is constant. (b) Find the equilibrium points. Is there a disease-free equilibria (I = 0)? Is there an endemic equilibria (I = 0)? If so, when does it exist? (c) Determine the local stability of each of the equilibrium points you found in (b). 27. Here is another approach to the analysis of the SIR model (2.11). (a) Divide the two equations of the model to give I′ (βS − α)I α dI = = −1 + . = S′ dS −βSI βS (b) Integrate to find the orbits in the (S, I )-plane, I = −S +

α logS + c, β

with c an arbitrary constant of integration. (c) Define the function V (S, I ) = S + I −

α log S β

and show that each orbit is given implicitly by the equation V (S, I ) = c for some choice of the constant c.

60

2 Simple Compartmental Models for Disease Transmission

(d) Show that no orbit reaches the I -axis and deduce that S∞ = limt→∞ S(t) > 0, which implies that part of the population escapes infection. 28. For the model (2.16) show that the final total population size is given by N∞ =

α d N0 + S∞ . α+d α+d

29. Consider the basic SIR model with disease deaths, but we now consider a vaccination class in place of recovery: S ′ (t) = μN − βS NI − (μ + φ)S I ′ (t) = βS NI − (μ + γ + δ)I V ′ (t) = γ I + φS − μV .

(2.26)

(a) Let R0 (φ) be R0 when φ = 0, and R0 (0) be R0 when φ = 0. Compute R0 (φ). What is the value of R0 (0)? Compare R0 (φ) with R0 (0). (b) Compute the equilibria. (c) Do the local stability analysis of the disease-free equilibrium state and the endemic state.

References 1. Agur,Z.L., G. Mazor, R. Anderson, and Y. Danon (1993) Pulse mass measles vaccination across age cohorts, Proc. Nat. Acad. Sci., 90:11698–11702. 2. Anderson, R.M. & R.M. May (1991) Infectious Diseases of Humans. Oxford University Press (1991) 3. Arreola R., A. Crossa, and M.C. Velasco (2000) Discrete-time SEIS models with exogenous re-infection and dispersal between two patches, Department of Biometrics, Cornell University, Technical Report Series, BU-1533-M. 4. Castillo-Chavez, C., K. Cooke, W. Huang, and S.A. Levin (1989a) The role of long incubation periods in the dynamics of HIV/AIDS. Part 1: Single Populations Models, J. Math. Biol., 27:373-98. 5. Castillo-Chavez, C., W. Huang, and J. Li (1996a) Competitive exclusion in gonorrhea models and other sexually-transmitted diseases, SIAM J. Appl. Math, 56: 494–508. 6. Castillo-Chavez, C., W. Huang, and J. Li (1997) The effects of females’ susceptibility on the coexistence of multiple pathogen strains of sexually-transmitted diseases, Journal of Mathematical Biology, 35:503–522. 7. Castillo-Chavez C., and A.A. Yakubu (2000) Epidemics models on attractors, Contemporary Mathematics, AMS, 284: 23–42. John Wiley & Sons, New York, 2000. 8. Dietz, K. (1982) Overall patterns in the transmission cycle of infectious disease agents. In: R.M. Anderson, R.M. May (eds) Population Biology of Infectious Diseases. Life Sciences Research Report 25, Springer-Verlag, Berlin-Heidelberg-New York, pp. 87–102 (1982)

References

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9. Engbert, R. and F. Drepper (1994) Chance and chaos in population biology-models of recurrent epidemics and food chain dynamics, Chaos, Solutions & Fractals, 4(7):1147–1169. 10. Evans, A.S. (1982) Viral Infections of Humans, 2nd ed., Plenum Press, New York. 11. Heesterbeek, J.A.P. and J.A.J Metz (1993) The saturating contact rate in marriage and epidemic models. J. Math. Biol., 31: 529–539. 12. Hethcote, H.W. (1976) Qualitative analysis for communicable disease models, Math. Biosc., 28:335–356. 13. Hethcote, H.W. (1989) Three basic epidemiological models. In Applied Mathematical Ecology, S. A. Levin, T.G. Hallam and L.J. Gross, eds., Biomathematics 18, 119–144, Springer-Verlag, Berlin-Heidelberg-New York. 14. Hethcote, H.W. and S.A. Levin (1989) Periodicity in epidemic models. In : S.A. Levin, T.G. Hallam & L.J. Gross (eds) Applied Mathematical Ecology, Biomathematics 18, SpringerVerlag, Berlin-Heidelberg-New York: pp. 193–211. 15. Hethcote, H.W., H.W. Stech and P. van den Driessche (1981) Periodicity and stability in epidemic models: a survey. In: S. Busenberg & K.L. Cooke (eds.) Differential Equations and Applications in Ecology, Epidemics and Population Problems, Academic Press, New York: 65–82. 16. Hurwitz, A. (1895) Über die Bedingungen unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen bezizt, Math. Annalen 46: 273–284. 17. Kermack, W.O. & A.G. McKendrick (1927) A contribution to the mathematical theory of epidemics. Proc. Royal Soc. London. 115: 700–721. 18. Kermack, W.O. & A.G. McKendrick (1932) Contributions to the mathematical theory of epidemics, part. II. Proc. Roy. Soc. London, 138: 55–83. 19. Kermack, W.O. & A.G. McKendrick (1932) Contributions to the mathematical theory of epidemics, part. III. Proc. Roy. Soc. London, 141: 94–112. 20. Mena-Lorca, J. & H.W. Hethcote (1992) Dynamic models of infectious diseases as regulators of population size. J. Math. Biol., 30: 693–716. 21. Raggett, G.F. (1982) Modeling the Eyam plague, IMA Journal, 18:221–226. 22. Routh, E.J. (1877) A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion, MacMillan. 23. Sánchez B.N., P.A. Gonzalez, R.A. Saenz (2000) The influence of dispersal between two patches on the dynamics of a disease, Department of Biometrics, Cornell University, Technical Report Series, BU-1531-M. 24. Shulgin, B., L. Stone, and Z. Agur (1998) Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol., 60:1123–1148. 25. Stone, L., B. Shulgin, Z. Agur (2000) Theoretical examination of the pulse vaccination in the SIR epidemic model, Math. and Computer Modeling, 31:207–215.

Chapter 3

Endemic Disease Models

In this chapter, we consider models for disease that may be endemic. In the preceding chapter we studied SI S models with and without demographics and SI R models with demographics. In each model, the basic reproduction number R0 determined a threshold. If R0 < 1 the disease dies out, while if R0 > 1 the disease becomes endemic. The analysis in each case involves determination of equilibria and determining the asymptotic stability of each equilibrium by linearization about the equilibrium. In each of the cases studied in the preceding chapter the disease-free equilibrium was asymptotically stable if and only if R0 < 1 and if R0 > 1 there was a unique endemic equilibrium that was asymptotically stable. In this chapter, we will see that these properties continue to hold for many more general models, but there are situations in which there may be an asymptotically stable endemic equilibrium when R0 < 1, and other situations in which there is an endemic equilibrium that is unstable for some values of R0 > 1. In Sect. 2.3 we analyzed the SI R model for diseases from which infectives recover with immunity against reinfection: S ′ = Λ(N ) − βSI − μS

I ′ = βSI − μI − αI − dI

(3.1)

N ′ = Λ(N ) − dI − μN. The following basic result holds for (3.1).

Theorem 3.1 The basic reproduction number for the model (3.1) is given by R0 =

K βK = . μ+α S∞

If R0 < 1, the system has only the disease-free equilibrium and this equilibrium is asymptotically stable. © Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_3

63

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3 Endemic Disease Models

Here, K is the population carrying capacity and S∞ is the susceptible population size at the endemic equilibrium. The theorem says that the disease-free equilibrium is locally asymptotically stable. We recall that this means that solutions with initial values close to this equilibrium remain close to the equilibrium and approach the equilibrium as t → ∞. In fact, it is not difficult to prove that this asymptotic stability is global, that is, that every solution approaches the disease-free equilibrium. If the quantity R0 is greater than one, then the disease-free equilibrium is unstable, but there is an endemic equilibrium that is (locally) asymptotically stable. In fact, these properties hold for some endemic disease models with more complicated compartmental structure . We will describe some examples.

3.1 More Complicated Endemic Disease Models 3.1.1 Exposed Periods In many infectious diseases there is an exposed period after the transmission of infection from susceptibles to potentially infective members but before these potential infectives develop symptoms and can transmit infection. To incorporate an exposed compartment with mean exposed period 1/κ we add an exposed class E and use compartments S, E, I, R and total population size N = S + E + I + R to give a generalization of the epidemic model (3.1) S ′ = Λ(N ) − βSI − μS

E ′ = βSI − (κ + μ)E

(3.2)

′

I = κE − (α + μ)I. A flow chart is shown in Fig. 3.1. The analysis of this model is similar to the analysis of (3.1), but with I replaced by E + I . That is, instead of using the number of infectives as one of the variables we use the total number of infected members, whether or not they are capable of transmitting infection.

Fig. 3.1 Flow chart for the SEI R endemic model (3.2)

3.1 More Complicated Endemic Disease Models

65

3.1.2 A Treatment Model One form of treatment that is possible for some diseases is vaccination to protect against infection before the beginning of an epidemic. For example, this approach is commonly used for protection against annual influenza outbreaks. A simple way to model this would be to reduce the total population size by the fraction of the population protected against infection. In reality, such inoculations are only partly effective, decreasing the rate of infection and also decreasing infectivity if a vaccinated person does become infected. This may be modeled by dividing the population into two groups with different model parameters which would require some assumptions about the mixing between the two groups. This is not difficult but we will not explore this direction until Chap. 5 on heterogeneous mixing. If there is a treatment for infection once a person has been infected, this may be modeled by supposing that there is a rate γ proportional to the number of infectives at which infectives are selected for treatment, and that treatment reduces infectivity by a fraction δ. Suppose that the rate of removal from the treated class is η. This leads to the SI T R model, where T is the treatment class, given by S ′ = μN − βS[I + δT ] − μS

I ′ = βS[I + δT ] − (α + γ + μ)I

(3.3)

′

T = γ I − (η + μ)T . A flow chart is shown in Fig. 3.2. In this model, we assume that the natural birth and death rates are equal so that the total population size remains constant. In order to calculate the basic reproduction number, we observe that an infective in a totally susceptible population causes βN new infections in unit time, and the

Fig. 3.2 Flow chart for the SI T R endemic model (3.3)

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3 Endemic Disease Models

mean time spent in the infective compartment is 1/(α + γ + μ). In addition, a fraction γ /(α + γ + μ) of infectives are treated because (γ + α + μ) is the rate at which the number of infectives decreases overall while γ is the rate at which these infectives are selected for treatment. While in the treatment stage the number of new infections caused in unit time is δβN, and the mean time in the treatment class is 1/(η + μ). Thus R0 =

γ δβN βN + . α+γ +μ α+γ +μη+μ

(3.4)

It is possible that if δ < 1 and α > η the treatment may increase the reproduction number. However, since α > η would mean that treatment prolongs the infection, this is quite unlikely. The equilibrium conditions for the model (3.3) are μN = βS[I + δT ] + μS βS[I + δT ] = (α + γ + μ)I

(3.5)

γ I = (η + μ)T . Substitution of the last of these equilibrium conditions into the second gives βS

δγ + η + μ I = (α + γ + μ)I, η+μ

and this implies that either I = 0 (disease-free equilibrium) or βS =

(α + γ + μ)(η + μ) δγ + η + μ

(endemic equilibrium). An endemic equilibrium exists if and only if the value of S given by this condition is less than N, and this is equivalent to R0 > 1. The Jacobian matrix or matrix of the linearization of (3.3) at an equilibrium (S, I, T ) is ⎡ ⎤ −β(I + δT ) − μ −βS −δβS ⎣ β(I + δT ) βS − (α + γ + μ) δβS ⎦ , 0 γ −(η + μ) and at the disease-free equilibrium (N, 0, 0) this is ⎡ ⎤ −μ −βN −δβN ⎣ 0 βN − (α + γ + μ) δβN ⎦ . 0 γ −(η + μ)

3.1 More Complicated Endemic Disease Models

67

The eigenvalues of this matrix are −μ and the eigenvalues of the 2 × 2 matrix   βN − (α + γ + μ) δβN . γ −(η + μ)

The eigenvalues of a 2 × 2 matrix have negative real part if and only if the matrix has negative trace and positive determinant. The condition that the determinant is positive is βN
1, we must make use of the four conditions [26, 37] introduced in Chap. 2. A somewhat complicated calculation shows that this is indeed the case.

3.1.3 Vertical Transmission In some diseases, notably Chagas’ disease, HIV/AIDS, hepatitis B, and rinderpest (in cattle), infection may be transferred not only horizontally (by contact between individuals) but also vertically (from an infected parent to a newly born offspring) [8]. We formulate an SI R model with vertical transmission by assuming that a fraction q of the offspring of infective members of the population are infective at birth. For simplicity, we assume that there are no disease deaths so that the total population size N is constant, and our model is based on (3.1). The birth rate in this model is Λ = μN, and we assume that births are distributed proportionally among compartments. Thus the rate of births to infectives is μI , the rate of newborn infectives is qμI , and the rate of newborn susceptibles is μN − qμI . This leads to the model S ′ = μN − qμI − βSI − μS I ′ = qμI + βSI − μI − αI.

(3.8)

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3 Endemic Disease Models

From the second equation, we see that equilibrium requires either I = 0 (diseasefree) or βS = μ(1 − q) + α. At the disease-free equilibrium, S = N, I = 0, and the matrix of the linearization is   −μ −qμ − βN . 0 βN − μ(1 − q) − α Thus the disease-free equilibrium is asymptotically stable if and only if βN < μ(1 − q) + α. This suggests that R0 =

βN + μq . μ+α

To see that this is indeed correct, we note that the term βN/(μ + α) represents horizontally transmitted infections at rate βN over a death-adjusted infective period μq 1/(μ + α), and the term μ+α represents vertically transmitted infections per infective. It is not difficult to verify that the endemic equilibrium, which exists if and only if R0 > 1 is asymptotically stable.

3.2 Some Applications of the SI R Model 3.2.1 Herd Immunity In order to prevent a disease from becoming endemic, it is necessary to reduce the basic reproduction number R0 below one. This may sometimes be achieved by immunization. If a fraction p of the Λ(N ) newborn members per unit time of the population is successfully immunized, the effect is to replace N by N (1 − p), and thus to reduce the basic reproduction number to R0 (1 − p). The requirement R0 (1 − p) < 1 gives 1 − p < 1/R0 , or p >1−

1 . R0

A population is said to have herd immunity if a large enough fraction has been immunized to assure that the disease cannot become endemic. The only disease for which this has actually been achieved worldwide is smallpox for which R0 is approximately 5, so that 80% immunization does provide herd immunity, and rinderpest, a cattle disease. For measles, epidemiological data in the USA indicate that R0 for rural populations ranges from 5.4 to 6.3, requiring vaccination of 81.5–84.1% of the

3.2 Some Applications of the SI R Model

69

population. In urban areas R0 ranges from 8.3 to 13.0, requiring vaccination of 88.0–92.3% of the population. In Great Britain, R0 ranges from 12.5 to 16.3, requiring vaccination of 92–94% of the population. The measles vaccine is not always effective, and vaccination campaigns are never able to reach everyone. As a result, herd immunity against measles has not been achieved (and probably never can be). An additional issue is that an anti-vaccination movement has developed, partly because of a fallacious belief that there is a link between the measles-mumpsrubella vaccine and the development of autism and partly because of a general opposition to vaccines. Since smallpox is viewed as more serious and requires a lower percentage of the population be immunized, herd immunity was attainable for smallpox. In fact, smallpox has been eliminated; the last known case was in Somalia in 1977, and the virus is maintained now only in laboratories. The eradication of smallpox was actually more difficult than expected because high vaccination rates were achieved in some countries but not everywhere, and the disease persisted in some countries. The eradication of smallpox was possible only after an intensive campaign for worldwide vaccination [22].

3.2.2 Age at Infection In order to calculate the basic reproduction number R0 for a disease modeled by a system (3.1), we need to know the values of the contact rate β and the parameters μ, K, and α. The parameters μ, K, and α can usually be measured experimentally but the contact rate β is difficult to determine directly. There is an indirect method of estimating R0 in terms of the life expectancy and the mean age at infection which enables us to avoid having to estimate the contact rate. In this calculation, we will assume that β is constant, but we will also indicate the modifications needed when β is a function of total population size N. The calculation assumes exponentially distributed life spans and infective periods. The result is valid so long as the life span is exponentially distributed, but if the life span is not exponentially distributed the result could be quite different. Consider the “age cohort” of members of a population born at some time t0 and let a be the age of members of this cohort. If y(a) represents the fraction of members of the cohort who survive to age (at least) a, then the assumption that a fraction μ of the population dies per unit time means that y ′ (a) = −μy(a). Since y(t0 ) = 1, we may solve this first order initial value problem to obtain y(a) = e−μa . The fraction dying at (exactly) age a is −y ′ (a) = μy(a). The mean life span is the average age ∞ at death, which is 0 a[−y ′ (a)]da, and if we integrate by parts we find that this life expectancy is 

∞

t0

[−ay ′ (a)] da = [−ay(a)]∞ t0 +



∞ t0

y(a) da =



∞ t0

y(a) da.

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3 Endemic Disease Models

Since y(a) = e−μa , this reduces to 1/μ. The life expectancy is often denoted by L, so that we may write L=

1 . μ

The rate at which surviving susceptible members of this cohort become infected at age a and time t0 + a is βI (t0 + a). Thus, if z(a) is the fraction of the age cohort alive and still susceptible at age a, z′ (a) = −[μ + βI (t0 + a)]z(a). Solution of this first linear order differential equation gives a

z(a) = e−[μa+

0

βI (t0 +b) db]

= y(a)e−

a 0

βI (t0 +b) db

.

The mean length of time in the susceptible class for members who may become infected, as opposed to dying while still susceptible, is 

∞

e−

0

a 0

βI (t0 +b)db

da,

and this is the mean age at which members become infected. If the system is at an equilibrium I∞ , this integral may be evaluated, and the mean age at infection, denoted by A, is given by A=



0

∞

e−βI∞ a da =

1 . βI∞

For our model the endemic equilibrium is I∞ =

μ μK − , μ+α β

and this implies βI∞ L = = R0 − 1. A μ

(3.9)

This relation is very useful in estimating basic reproduction numbers. For example, in some urban communities in England and Wales between 1956 and 1969 the average age of contracting measles was 4.8 years. If life expectancy is assumed to be 70 years, this indicates R0 = 15.6. If β is a function β(N) of total population size and K is the carrying capacity, the relation (3.9) becomes   L β(K) R0 = 1+ . β(N0 ) A

3.2 Some Applications of the SI R Model

71

If disease mortality does not have a large effect on total population size, in particular if there is no disease mortality, this relation is very close to (3.9). The relation between age at infection and basic reproduction number indicates that measures such as inoculations, which reduce R0 , will increase the average age at infection. For diseases such as rubella (German measles), whose effects may be much more serious in adults than in children, this indicates a danger that must be taken into account: While inoculation of children will decrease the number of cases of illness, it will tend to increase the danger to those who are not inoculated or for whom the inoculation is not successful. Nevertheless, the number of infections in older people will be reduced, although the fraction of cases which are in older people will increase.

3.2.3 The Inter-Epidemic Period Many common childhood diseases, such as measles, whooping cough, chicken pox, diphtheria, and rubella, exhibit variations from year to year in the number of cases. These fluctuations are frequently regular oscillations, suggesting that the solutions of a model might be periodic. This does not agree with the predictions of the model we have been using in this section; however, it would not be inconsistent with solutions of the characteristic equation, which are complex conjugate with small negative real part corresponding to lightly damped oscillations approaching the endemic equilibrium. Such behavior would look like recurring epidemics. If the eigenvalues of the matrix of the linearization at an endemic equilibrium are −u ± iv, where i 2 = −1, then the solutions of the linearization are of the form Be−ut cos(vt + c), with decreasing “amplitude” Be−ut and “period” 2π v . For the model (3.1) we recall that at the endemic equilibrium we have βI∞ + μ = μR0 ,

βS∞ = μ + α

and the matrix of the linearization is   −μR 0 −(μ + α) . 0 μ(R0 − 1) The eigenvalues are the roots of the quadratic equation λ2 + μR0 λ + μ(R0 − 1)(μ + α) = 0, which are

λ=

−μR0 ±

 μ2 R0 2 − 4μ(R0 − 1)(μ + α) 2

.

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3 Endemic Disease Models

If the mean infective period 1/α is much shorter than the mean life span 1/μ, we may neglect the terms that are quadratic in μ. Thus, the eigenvalues are approximately √ −4μ(R0 − 1)α , 2 √ and these are complex with imaginary part μ(R0 − 1)α. This indicates oscillations with period approximately −μR0 ±

√

2π . μ(R0 − 1)α

We use the relation μ(R0 − 1) = μL/A and the mean infective period τ = 1/α to √ see that the interepidemic period T is approximately 2π Aτ . Thus, for example, for recurring outbreaks of measles with an infective period of 2 weeks or 1/26 year in a population with a life expectancy of 70 years with R0 estimated as 15, we would expect outbreaks spaced 2.76 years apart. Also, as the “amplitude” at time t is e−μR0 t/2 , the maximum displacement from equilibrium is multiplied by a factor e−(15)(2.76)/140 = 0.744 over each cycle. In fact, many observations of measles outbreaks indicate less damping of the oscillations, suggesting that there may be additional influences that are not included in our simple model. To explain oscillations about the endemic equilibrium a more complicated model is needed. One possible generalization would be to assume seasonal variations in the contact rate [13, 27]. This is a reasonable supposition for a childhood disease most commonly transmitted through school contacts, especially in winter in cold climates. Note, however, that data from observations are never as smooth as model predictions and models are inevitably gross simplifications of reality which cannot account for random variations in the variables. It may be difficult to judge from experimental data whether an oscillation is damped or persistent.

3.2.4 “Epidemic” Approach to Endemic Equilibrium In the model (3.1) the demographic time scale described by the birth and natural death rates μK and μ and the epidemiological time scale described by the rate α of departure from the infective class may differ substantially. Think, for example, of a natural death rate μ = 1/75, corresponding to a human life expectancy of 75 years, and epidemiological parameter α = 25, describing a disease from which all infectives recover after a mean infective period of 1/25 year, or 2 weeks. Suppose we consider a carrying capacity K = 1000 and take β = 0.1, indicating that an average infective makes (0.1)(1000)= 100 contacts per year. Then R0 = 4.00, and at the endemic equilibrium we have S∞ = 250.13, I∞ = 0.40, R∞ = 749.47. This

3.3 Temporary Immunity

73

equilibrium is globally asymptotically stable and is approached from every initial state. However, if we take S(0) = 999, I (0) = 1, R(0) = 0, simulating the introduction of a single infective into a susceptible population and solve the system numerically we find that the number of infectives rises sharply to a maximum of 400 and then decreases to almost zero in a period of 0.4 year, or about 5 months. In this time interval the susceptible population decreases to 22 and then begins to increase, while the removed (recovered and immune against reinfection) population increases to almost 1000 and then begins a gradual decrease. The size of this initial “epidemic” could not have been predicted from our qualitative analysis of the system (3.1). On the other hand, since μ is so small compared to the other parameters of the model, we might consider neglecting μ, replacing it by zero in the model. If we do this, the model reduces to the simple Kermack–McKendrick epidemic model (without births and deaths) of Sect. 2.4. If we follow the model (3.1) over a longer time interval we find that the susceptible population grows to 450 after 46 years, then drops to 120 during a small epidemic with a maximum of 18 infectives, and exhibits widely spaced epidemics decreasing in size. It takes a very long time before the system comes close to the endemic equilibrium and remains close to it. The large initial epidemic conforms to what has often been observed in practice when an infection is introduced into a population with no immunity, such as the smallpox inflicted on the Aztecs by the invasion of Cortez. If we use the model (3.1) with the same values of β, K, and μ, but take α = 0, d = 25 to describe a disease fatal to all infectives, we obtain very similar results. Now the total population is S + I , which decreases from an initial size of 1000 to a minimum of 22 and then gradually increases and eventually approaches its equilibrium size of 250.53. Thus, the disease reduces the total population size to one-fourth of its original value, suggesting that infectious diseases may have large effects on population size. This is true even for populations which would grow rapidly in the absence of infection, as we shall see in a later section (Sect. 3.7).

3.3 Temporary Immunity In the SI R models that we have studied, it has been assumed that the immunity received by recovery from the disease is permanent. This is not always true, as there may be a gradual loss of immunity with time. In addition, there are often mutations in a virus, and as a result the active disease strain is sufficiently different from the strain from which an individual has recovered and the immunity received may wane. Temporary immunity may be described by an SI RS model in which a rate of transfer from R to S is added to an SI R model. For simplicity, we confine our attention to epidemic models, without including births, natural deaths, and disease deaths, but the analysis of models including births and deaths would lead to the same conclusions. Thus we begin with a model

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3 Endemic Disease Models

S ′ = −βSI + θ R I ′ = βSI − αI

R ′ = αI − θ R,

with a proportional rate θ of loss of immunity. Since N ′ = (S + I + R)′ = 0, the total population size N is constant, and we may replace R by N − S − I and reduce the model to a two-dimensional system S ′ = −βSI + θ (N − S − I ) I ′ = βSI − αI.

(3.10)

Equilibria are solutions of the system βSI + θ S + θ I = θ N αI + θ S + θ I = θ N, and there is a disease-free equilibrium S = α/β, I = 0. If R0 = βN/α > 1, there is also an endemic equilibrium with βS = α,

(α + θ )I = θ (N − S).

The matrix of the linearization of (3.10) at an equilibrium (S, I ) is A=

  −(βI + θ ) −(βS + θ ) . βI βS − α

At the disease-free equilibrium A has the sign structure   − − . 0 βN − α This matrix has negative trace and positive determinant if and only if βN < α, or R0 < 1. At an endemic equilibrium, the matrix has sign structure   − − . + 0 and thus always has negative trace and positive determinant. We see from this that, as in other models studied in this chapter, the disease-free equilibrium is asymptotically stable if and only if the basic reproduction number is less than 1 and the endemic equilibrium, which exists if and only if the basic reproduction number

3.3 Temporary Immunity

75

exceeds 1, is always asymptotically stable. However, it is possible for a different SI RS model to have quite different behavior.

3.3.1 *Delay in an SI RS Model We consider an SI RS model, which assumes a constant period of temporary immunity following recovery from the infection in place of an exponentially distributed period of temporary immunity [24]. We assume that there is a temporary immunity period of fixed length ω, after which recovered infectives revert to the susceptible class. The resulting model is described by the system of differential– difference equations S ′ (t) = −βS(t)I (t) + αI (t − ω) I ′ (t) = βS(t)I (t) − αI (t)

(3.11)

R ′ (t) = αI (t) − αI (t − ω). The equilibrium analysis of a system of differential–difference equations with a delay ω is analogous to the equilibrium analysis of a system of ordinary differential equations, but there are important variations. Instead of assigning an initial condition at t = 0 it is necessary to assign initial data on the interval −ω ≤ t ≤ 0. Equilibria of a system of differential–difference equations are constant solutions, just as for systems of differential equations, and the process of linearization about an equilibrium is the same. The characteristic equation at an equilibrium is the condition that the linearization at the equilibrium has a solution whose components are constant multiples of eλt . In the ordinary differential equation case, this is just the equation that determines the eigenvalues of the coefficient matrix, a polynomial equation, but in the general case, it is a transcendental equation. The result on which our analysis depends, which we state without proof, is that an equilibrium is asymptotically stable if all roots of the characteristic equation have negative real part, or equivalently that the characteristic equation have no roots with real part greater than or equal to zero [5]. In (3.11), since N = S + I + R is constant, we may discard the equation for R and use a two-dimensional model S ′ (t) = −βS(t)I (t) + αI (t − ω) I ′ (t) = βS(t)I (t) − αI (t).

(3.12)

Equilibria are given by I = 0 or βS = α. There is a disease-free equilibrium S = N, I = 0. There is also an endemic equilibrium for which βS = α. However, the two equations for S and I give only a single equilibrium condition. To determine the

76

3 Endemic Disease Models

endemic equilibrium (S∞ , I∞ ), we must write the equation for R in the integrated form  t R(t) = αI (x)dx t−ω

to give R∞ = ωαI∞ . We also have βS∞ = α, and from S∞ + I∞ + R∞ = N we obtain βI∞ =

(βN − α) . 1 + ωα

To linearize about an equilibrium (S∞ , I∞ ) of (3.12) we substitute S(t) = S∞ + u(t),

I (t) = I∞ + v(t),

and neglect the quadratic term, giving the linearization u′ (t) = −βI∞ u(t) − βS∞ v(t) + αv(t − ω) v ′ (t) = βI∞ u(t) + βS∞ v(t) − αv(t).

The characteristic equation is the condition on λ that this linearization has a solution u(t) = u0 eλt ,

v(t) = v0 eλt ,

and this is (βI∞ + λ)u0 + (βS∞ − αe−λω )v0 = 0 βI∞ u0 + (βS∞ − α − λ)v0 = 0, or  λ + βI∞ βS∞ − αeλω . det βI∞ βS∞ − α − λ 

This reduces to αβI∞

1 − e−ωλ = −[λ + α + βS∞ + βI∞ ]. λ

(3.13)

At the disease-free equilibrium S∞ = N, I∞ = 0, this reduces to a linear equation with a single root λ = −βN − α, which is negative if and only if R0 = βN/α < 1. We think of ω and N as fixed and consider β and α as parameters. If α = 0 the Eq. (3.13) is linear and its only root is −βS∞ − βI∞ < 0. Thus, there is a region in

3.3 Temporary Immunity

77

the (α, β) parameter space containing the β-axis, in which all roots of (3.13) have negative real part. In order to find how large this stability region is, we make use of the fact that the roots of (3.13) depend continuously on β and α. A root can move into the right half-plane only by passing through the value zero or by crossing the imaginary axis as βN and α vary. Thus, the stability region contains the β-axis and extends into the plane until there is a root λ = 0 or until there is a pair of pure imaginary roots λ = ±iy with y > 0. Since the left side and right side of (3.13) have opposite sign for real λ ≥ 0, there cannot be a root λ = 0. The condition that there is a root λ = iy is αβI∞

1 − e−iωα = −(iy + α + βS∞ + βI∞ ) iy

(3.14)

and separation into real and imaginary parts gives the pair of equations αβ

sin ωy = −[α + βS∞ + βI∞ ], y

αβI∞

1 − cos ωy = y. y

(3.15)

To satisfy the first condition, it is necessary to have ωα > 1 since | sin ωy| ≤ |ωy| for all y. This implies, in particular, that the endemic equilibrium is asymptotically stable if ωα < 1. In addition, it is necessary to have sin ωy < 0. There is an infinite sequence of intervals on which sin ωy < 0, the first being π < ωy < 2π . For each of these intervals, the equations (3.15) define a curve in the (β, α) plane parametrically with y as parameter. The region in the plane below the first of these curves is the region of asymptotic stability, that is, the set of values of β and α for which the endemic equilibrium is asymptotically stable. This curve is shown for ω = 1, N = 1 in Fig. 3.3. Since R0 = βN/α > 1, only the portion of the (β, α) plane below the line α = βN is relevant. The new feature of the model of this section is that the endemic equilibrium is not asymptotically stable for all parameter values. What is the behavior of the model if the parameters are such that the endemic equilibrium is unstable? A plausible suggestion is that since the loss of stability corresponds to a root λ = iy of the characteristic equation there are solutions of the model behaving like the real part of eiyt , that is, that there are periodic solutions. This is exactly what does happen according to a very general result called the Hopf bifurcation theorem [25], which says that when roots of the characteristic equation cross the imaginary axis a stable periodic orbit arises. From an epidemiological point of view periodic behavior is unpleasant. It implies fluctuations in the number of infectives which makes it difficult to allocate resources for treatment. It is also possible for oscillations to have a long period. This means that if data are measured over only a small time interval the actual behavior may not be displayed. Thus, the identification of situations in which an endemic equilibrium is unstable is an important problem.

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3 Endemic Disease Models

Fig. 3.3 Region of asymptotic stability for endemic equilibria (ω = 1, N = 1)

3.4 A Simple Model with Multiple Endemic Equilibria In compartmental models for the transmission of communicable diseases there is usually a basic reproduction number R0 , representing the mean number of secondary infections caused by a single infective introduced into a susceptible population. If R0 < 1 there is a disease-free equilibrium which is asymptotically stable, and the infection dies out. If R0 > 1 the usual situation is that there is a unique endemic equilibrium which is asymptotically stable, and the infection persists. Even if the endemic equilibrium is unstable, the instability commonly arises from a Hopf bifurcation [25], described in Sect. 3.3, and the infection still persists but in an oscillatory manner. More precisely, as R0 increases through 1 there is an exchange of stability between the disease-free equilibrium and the endemic equilibrium (which is negative as well as unstable and thus biologically meaningless if R0 < 1). There are, however, situations in which there may be more than one endemic equilibrium even in very simple epidemic models, and we describe such a model suggested in [42, 43]. We consider an SI S model in a population of constant total size N with treatment of infectives, assuming that the treatment cures the infection but that there is a maximum capacity for treatment. Thus we assume a model

3.4 A Simple Model with Multiple Endemic Equilibria

S ′ = −βSI + h(I ) I ′ = βSI − αI − h(I ),

79

(3.16)

assuming a treatment function h(I ) of the form h(I ) =



rI, rI ∗ ,

(I < I ∗ ) (I ≥ I ∗ ),

in which r is a constant representing the treatment rate up to a maximum capacity rI ∗ . Since the total population size S + I is a constant N, we may replace S by N − I and reduce the model to a single equation I ′ = βI (N − I ) − αI − h(I ) = g(I ).

(3.17)

There is a disease-free equilibrium I = 0, and it is easily verified that the diseasefree equilibrium is asymptotically stable if and only if R0 = βN/(α + r) < 1. For I ≤ I ∗ , g(I ) = βI (N − I ) − (α + r)I, and an endemic equilibrium with I ≤ I ∗ is a positive solution I∞ of g(I ) = 0, namely I∞ = N −

  1 α+r , =N 1− β R0

and there is such an equilibrium if and only if   α+r 1 I ≥N− . =N 1− β R0 ∗

(3.18)

For I ≤ I ∗ , g ′ (I ) = β(N − 2βI − (α + r), and g ′ (I∞ ) < 0 if and only if

  α+r α+r , < 2I∞ = 2 N − N− β β

and this is equivalent to R0 > 1. Thus, the equilibrium I∞ ≤ I ∗ exists and is asymptotically stable if and only if (3.18) is satisfied.

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3 Endemic Disease Models

Equilibria I > I ∗ are solutions of the quadratic equation g(I ) = −βI 2 + (βN − α)I − rI ∗ = 0, which are I=

(βN −α) +



(βN−α)2 −4rβI ∗ , 2β

J =

(βN −α) −

 (βN − α)2 −4βrI ∗ . 2β

Then J < (βN − α)/2β and I > (βN − α)/2β. For these to qualify as equilibria, they must also be greater than I ∗ and less than N, but it is possible to choose parameter values such that the model (3.16) has more than one endemic equilibrium. For example, the choices α = 0.5,

r = 0.5,

N = 1,

I ∗ = 0.05,

so that R0 = β, give two equilibria I, J for some values of β, including some values with R0 < 1. With these parameter values, I = J = 0.279 when β = 0.779. An equilibrium I∞ of the differential equation I ′ = g(I ) is asymptotically stable if g ′ (I∞ ) < 0, and unstable if g ′ (I∞ ) > 0. From this, it is easy to deduce that the equilibrium J is unstable, while the equilibrium I is asymptotically stable. If we plot the equilibrium values as functions of β, the curve I begins at the point (0.779, 0.279) and goes upwards to the right, while the curve J goes downward to the right from the same starting point. Because of the choice I ∗ = 0.05, only the portion of the J curve above the line I = 0.05 is relevant. For 0.779 ≤ R0 ≤ 1 there are two asymptotically stable equilibria, namely 0 and I separated by an unstable equilibrium J . For this reason, we have drawn the J curve as a dotted curve in Fig. 3.4.

Fig. 3.4 Multiple endemic equilibria

3.5 A Vaccination Model: Backward Bifurcations

81

A bifurcation curve, a graph of equilibria as a function of the basic reproduction number, as in Fig. 3.4, gives a good deal of information about the behavior of endemic equilibria. We observe, for example, that in Fig. 3.4, there are endemic equilibria for some values of the basic reproduction number less than 1, and that there is a discontinuity in the endemic equilibria at R0 = 1.

3.5 A Vaccination Model: Backward Bifurcations In a compartmental model, there is a bifurcation, or change in equilibrium behavior, at R0 = 1 but the equilibrium infective population size depends continuously on R0 . Such a transition is called a forward, or transcritical, bifurcation. The behavior at a bifurcation may be described graphically by the bifurcation curve, which is the graph of equilibrium infective population size I as a function of the basic reproduction number R0 . For a forward bifurcation, the bifurcation curve is as shown in Fig. 3.5. It has been noted [14, 20, 21, 29] that in epidemic models with multiple groups and asymmetry between groups or multiple interaction mechanisms it is possible to have a very different bifurcation behavior at R0 = 1. There may be multiple positive endemic equilibria for values of R0 < 1 and a backward bifurcation at R0 = 1. This means that the bifurcation curve has the form shown in Fig. 3.4 with a broken curve denoting an unstable endemic equilibrium that separates the domains of attraction of asymptotically stable equilibria. The qualitative behavior of an epidemic system with a backward bifurcation differs from that of a system with a forward bifurcation in at least three important ways. If there is a forward bifurcation at R0 = 1 it is not possible for a disease to invade a population if R0 < 1 because the system will return to the disease-free equilibrium I = 0 if some infectives are introduced into the population. On the other hand, if there is a backward bifurcation at R0 = 1 and enough infectives are Fig. 3.5 Forward bifurcation

82

3 Endemic Disease Models

introduced into the population to put the initial state of the system above the unstable endemic equilibrium with R0 < 1, the system will approach the asymptotically stable endemic equilibrium. Other differences are observed if the parameters of the system change to produce a change in R0 . With a forward bifurcation at R0 = 1 the equilibrium infective population remains zero so long as R0 < 1 and then increases continuously as R0 increases. With a backward bifurcation at R0 = 1, there is an asymptotically stable disease-free equilibrium so long as R0 < 1 but there is also an asymptotically stable endemic equilibrium for some values of R0 < 1 and as R0 increases through 1 the infective population size jumps to the positive endemic equilibrium. In the other direction, if a disease is being controlled by means that decrease R0 it is sufficient to decrease R0 to 1 if there is a forward bifurcation at R0 = 1 but it is necessary to bring R0 well below 1 if there is a backward bifurcation. These behavior differences are important in planning how to control a disease; a backward bifurcation at R0 = 1 makes control more difficult. One control measure often used is the reduction of susceptibility to infection produced by vaccination. By vaccination, we mean either an inoculation that reduces susceptibility to infection or an education program such as encouragement of better hygiene or avoidance of risky behavior for sexually transmitted diseases. Whether vaccination is inoculation or education, typically it reaches only a fraction of the susceptible population and is not perfectly effective. In an apparent paradox, models with vaccination may exhibit backward bifurcations, making the behavior of the model more complicated than the corresponding model without vaccination. It has been argued [6] that a partially effective vaccination program applied to only part of the population at risk may increase the severity of outbreaks of such diseases as HIV/AIDS. We will give a qualitative analysis of a model which may have a variable total population size N ≤ K for which there is a possibility of a backward bifurcation. The model we will study adds vaccination to the simple SIS model with births and natural deaths but with no disease deaths studied in Sect. 2.2. We have considered the model S ′ = Λ(N ) − β(N)SI − μS + αI I ′ = βSI − (μ + α)I,

(3.19)

where the population carrying capacity K is defined by Λ(K) = μK, Λ′ (K) < μ and the contact rate β(N) is a function of total population size with Nβ(N) nondecreasing and β(N) non-increasing. We have seen that there is a disease-free equilibrium I = 0 that is asymptotically stable if R0 =

Kβ(K) < 1. μ+α

If R0 > 1 the disease-free equilibrium is unstable but there is an endemic equilibrium that is asymptotically stable.

3.5 A Vaccination Model: Backward Bifurcations

83

To the model (3.19) we add the assumption that in unit time a fraction ϕ of the susceptible class is vaccinated. The vaccination may reduce but not completely eliminate susceptibility to infection. We model this by including a factor σ , 0 ≤ σ ≤ 1, in the infection rate of vaccinated members with σ = 0 meaning that the vaccine is perfectly effective and σ = 1 meaning that the vaccine has no effect. We describe the new model by including a vaccinated class V , with S ′ = μN − β(N)SI − (μ + ϕ)S + αI

I ′ = β(N)SI + σβ(N )V I − (μ + α)I

(3.20)

V ′ = ϕS − σβ(N )V I − μV

and N = S + I + V . Since N is constant, we can replace S by N − I − V to give the equivalent system I ′ = β [N − I − (1 − σ )V ] I − (μ + α)I V ′ = ϕ[N − I ] − σβV I − (μ + ϕ)V

(3.21)

with β = β(N). The system (3.21) is the basic vaccination model which we will analyze. We remark that if the vaccine is completely ineffective, σ = 1, then (3.21) is equivalent to an SI S model. If all susceptibles are vaccinated immediately (formally, ϕ → ∞), the model (3.21) is equivalent to I ′ = σβI (K − I ) − (μ + α)I which is an SI S model with basic reproduction number R0∗ =

σβK = σ R0 ≤ R 0 . μ+α

We will think of the parameters μ, α, ϕ, and σ as fixed and will view β as variable. In practice, the parameter ϕ is the one most easily controlled, and later we will express our results in terms of an uncontrolled model with parameters β, μ, α, and σ fixed and examine the effect of varying ϕ. With this interpretation in mind, we will use R(ϕ)to denote the basic reproduction number of the model (3.21), and we will see that R0∗ ≤ R(ϕ) ≤ R0 . Equilibria of the model (3.21) are solutions of βI [K − I − (1 − σ )V ] = (μ + α)I ϕ[K − I ] = σβV I + (μ + ϕ)V .

(3.22)

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3 Endemic Disease Models

If I = 0, then the first of these equations is satisfied and the second leads to V =

ϕ K. μ+ϕ

This is the disease-free equilibrium. The matrix of the linearization of (3.21) at an equilibrium (I, V ) is 

−2βI − (1 − σ )βV − (μ + α) + βK −(ϕ + σβV )

 −(1 − σ )βI . −(μ + ϕ + σβI )

At the disease-free equilibrium this matrix is 

−(1 − σ )βV − (μ + α) + βK −(ϕ + σβV )

0 −(μ + ϕ)



which has negative eigenvalues, implying the asymptotic stability of the disease-free equilibrium, if and only if −(1 − σ )βV − (μ + α) + βK < 0. Using the value of V at the disease-free equilibrium this condition is equivalent to R(ϕ) =

μ + σϕ βK μ + σϕ · = R0 < 1. μ+α μ+ϕ μ+ϕ

The case ϕ = 0 is that of no vaccination with R(0) = R0 , and R(ϕ) < R0 if ϕ > 0. We note that R0∗ = σ R0 = limϕ→∞ R(ϕ) < R0 . If 0 ≤ σ < 1 endemic equilibria are solutions of the pair of equations β [K − I − (1 − σ )V ] = μ + α ϕ[K − I ] = σβV I + (μ + ϕ)V .

(3.23)

We eliminate V using the first equation of (3.23) and substitute into the second equation to give an equation of the form AI 2 + BI + C = 0

(3.24)

with A = σβ B = (μ + θ + σ ϕ) + σ (μ + α) − σβK C=

(μ + α)(μ + θ + ϕ) − (μ + θ + σ ϕ)K. β

(3.25)

3.5 A Vaccination Model: Backward Bifurcations

85

If σ = 0 (3.24) is a linear equation with unique solution. I =K−

  1 (μ + α)(μ + ϕ) =K 1− βμ R(ϕ)

which is positive if and only if R(ϕ) > 1. Thus if σ = 0 there is a unique endemic equilibrium if R(ϕ) > 1 that approaches zero as R(ϕ) → 1+ and there cannot be an endemic equilibrium if R(ϕ) < 1. In this case, it is not possible to have a backward bifurcation at R(ϕ) = 1. We note that C < 0 if R(ϕ) > 1, C = 0 if R(ϕ) = 1, and C > 0 if R(ϕ) < 1. If σ > 0, so that (3.24) is quadratic and if R(ϕ) > 1 then there is a unique positive root of (3.24) and thus there is a unique endemic equilibrium. If R(ϕ) = 1, then C = 0 and there is a unique non-zero solution of (3.24) I = −B/A which is positive if and only if B < 0. If B < 0 when C = 0 there is a positive endemic equilibrium for R(ϕ) = 1. Since equilibria depend continuously on ϕ there must then be an interval to the left of R(ϕ) = 1 on which there are two positive equilibria I=

−B ±

√

B 2 − 4AC . 2A

This establishes that the system (3.21) has a backward bifurcation at R(ϕ) = 1 if and only if B < 0 when β is chosen to make C = 0. We can give an explicit criterion in terms of the parameters μ, ϕ, σ for the existence of a backward bifurcation at R(ϕ) = 1. When R(ϕ) = 1, C = 0 so that (μ + σ ϕ)βK = (μ + α)(μ + ϕ).

(3.26)

The condition B < 0 is (μ + σ ϕ) + σ (μ + α) < σβK with βK determined by (3.26), or σ (μ + α)(μ + ϕ) > (μ + σ ϕ) [(μ + σ ϕ) + σ (μ + α)] which reduces to σ (1 − σ )(μ + α)ϕ > (μ + σ ϕ)2 .

(3.27)

A backward bifurcation occurs at R(ϕ) = 1, with βK given by (3.26) if and only if (3.27) is satisfied. We point out that for an SI model, where α = 0, the condition (3.27) becomes σ (1 − σ )μϕ > (μ + σ ϕ)2 .

86

3 Endemic Disease Models

But (μ + σ ϕ)2 = μ2 + σ 2 ϕ 2 + 2μσ ϕ > 2μσ ϕ > σ (1 − σ )μϕ because σ < 1. Thus a backward bifurcation is not possible if α = 0, that is, for an SI model. Likewise, (3.27) cannot be satisfied if σ = 0. If C > 0 and either B ≥ 0 or B 2 < 4AC, there are no positive solutions of (3.24)and thus there are no endemic equilibria. Equation (3.24) has two positive solutions, corresponding to two endemic √ equilibria, if and only if√ C > 0, or R(ϕ) < 1, and B < 0, B 2 > 4AC, or B < −2 AC < 0. If B = −2 AC, there is one positive solution I = −B/2A of (3.24). If (3.27) is satisfied, so that there is a backward bifurcation at R(ϕ) = 1, there are two endemic equilibria for an interval of values of β from βK =

(μ + α)(μ + ϕ) μ + σϕ

√ corresponding to R(ϕ) = 1 to a value βc defined by B = −2 AC. To calculate βc , we let x = μ + α − βK, U = μ + σ ϕ to give B = σ x + U , βC = βKU + (μ + α)(μ + ϕ). Then B 2 = 4AC becomes (σ x + U )2 + 4βσ KU − 4σ (μ + α)(μ + ϕ) = 0 which reduces to  (σ x)2 − 2U (σ x) + U 2 + 4σ (1 − σ )(μ + α)ϕ = 0 with roots  σ x = U ± 2 σ (1 − σ )(μ + α)ϕ.

For the positive root B = σ x + U > 0, and since √ we require B < 0 as well as B 2 − 4AC = 0, we obtain βc from σ x = U − 2 σ (1 − σ )(μ + α)ϕ so that  σβc K = σ (μ + α) + 2 σ (1 − σ )(μ + α)ϕ − (μ + σ ϕ). (3.28) Then the critical basic reproduction number Rc is given by Rc =

√ μ + σ ϕ σ (μ + α) + 2 σ (1 − σ )(μ + α)ϕ − (μ + σ ϕ) · μ+ϕ σ (μ + α)ϕ

and it is possible to verify with the aid of (3.28) that Rc < 1.

3.5 A Vaccination Model: Backward Bifurcations

87

3.5.1 The Bifurcation Curve In drawing the bifurcation curve (the graph of I as a function of R(ϕ)), we think of β as variable with the other parameters μ, α, σ, Q, ϕ as constant. Then R(ϕ) is a constant multiple of β and we can think of β as the independent variable in the bifurcation curve. Implicit differentiation of the equilibrium condition (3.24) with respect to β gives (2AI + B)

(μ + α)(μ + ϕ) dI . = σ I (K − I ) + dβ β2

It is clear from the first equilibrium condition in (3.23) that I ≤ K and this implies that the bifurcation curve has positive slope at equilibrium values with 2AI + B > 0 and negative slope at equilibrium values with 2AI + B < 0. If there is not a backward bifurcation at R(ϕ) = 1, then the unique endemic equilibrium for R(ϕ) > 1 satisfies 2AI + B =

 B 2 − 4AC > 0

and the bifurcation curve has positive slope at all points where I > 0. Thus the bifurcation curve is as shown in Fig. 3.5. If there is a backward bifurcation at R(ϕ) = 1, then there is an interval on which there are two endemic equilibria given by  2AI + B = ± B 2 − 4AC.

The bifurcation curve has negative slope at the smaller of these and positive slope at the larger of these. Thus the bifurcation curve is as shown in Fig. 3.4. The condition 2AI + B > 0 is also significant in the local stability analysis of endemic equilibria. An endemic equilibrium of (3.21) is (locally) asymptotically stable if and only if it corresponds to a point on the bifurcation curve at which the curve is increasing. To prove this, we observe that the matrix of the linearization of (3.21) at an equilibrium (I, V ) is 

−2βI − (1 − σ )βV − (μ + α) + βK −(ϕ + σβV )

 −(1 − σ )βI . −(μ + ϕ + σβI )

Because of the equilibrium conditions (3.23), the matrix at an endemic equilibrium (I, V ) is 

−βI −(ϕ + σβV )

 −(1 − σ )βI . −(μ + ϕ + σβI )

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3 Endemic Disease Models

This has negative trace, and its determinant is σ (βI )2 + βI (μ + ϕ) − (1 − σ )ϕβI − (1 − σ )βV · σβI   = βI 2σβI + (μ + σ ϕ) + σ (μ + α) − σβK = βI [2AI + B].

If 2AI + B > 0, that is, if the bifurcation curve has positive slope, then the determinant is positive and the equilibrium is asymptotically stable. If 2AI +B < 0 the determinant is negative and the equilibrium is unstable. In fact, it is a saddle point. A saddle point in the plane is an equilibrium at which the linearization has one positive eigenvalue and one negative eigenvalue. This means that there are two orbits approaching the saddle point called stable separatrices and two orbits going out from the saddle point, called unstable separatrices. Because orbits cannot cross the separatrices, the stable separatrices divide the plane into two regions and divide the plane into two domains of attraction. The stable separatrices in the (I, V ) plane separate the domains of attraction of the other (asymptotically stable) endemic equilibrium and the disease-free equilibrium.

3.6 *An SEI R Model with General Disease Stage Distributions The ODE models considered in earlier parts of the chapter, except the SIRS model in Sect. 3.3, assume exclusively that the durations of disease stages (e.g., latent and infectious stages) are exponentially distributed. This assumption, while making the models and their analyses easier, is not biologically realistic for most infectious diseases. A more appropriate distribution is the gamma distribution, for which the probability of remaining in the stage is given by pn (s) =

n−1 (nθ s)k s −nθs k=0

k!

(3.29)

where 1/θ is the mean of the distribution and n is the shape parameter. The exponential distribution is the special case when n = 1. The other extreme case is when n → ∞, which corresponds to a fixed duration. Figure 3.6 illustrates the gamma distribution for various n values. Let 1/κ and 1/α denote the mean latent and infectious periods, and let m and n denote the shape parameter for the latent and infectious stages, respectively. It has been reported [44] that for measles 1/κ = 8, 1/α = 5, m = n = 20,

3.6 *An SEI R Model with General Disease Stage Distributions

89

Fig. 3.6 Depictions of the survival probability (3.29) for the infectious period (left) and the probability density function (right) with different shape parameter values (n). The special case of n = 1 gives the exponential distribution. The mean infectious period (1/θ) is chosen to be 1 week

and for smallpox 1/κ = 14, 1/α = 8.6, m = 40, n = 4. There are other cases where the disease stage durations do not fit well by the standard family distributions. Epidemiological models with non-exponential distributions such as the gamma distribution have been previously studied (see, for example, [23, 30, 31]. In these studies, the authors discussed various drawbacks associated with the exponential distribution assumption. For example, it is pointed out that constant recovery is a poor description of real-world infections, and they show that in models with more realistic distributions of disease stages less stable behavior may be expected and disease persistence may be diminished [23, 30]. In [18] it was demonstrated that when control measures such as quarantine and isolation are considered, models with exponential and gamma distributions may generate contradictory evaluations on control strategies. Thus, it will be helpful to have mathematical results for models that allow arbitrary distributions. This is the goal of this section. Let PE , PI : [0, ∞) → [0, 1] describe the durations of the exposed (latent) and infective stages, respectively. That is, Pi (s) (i = E, I ) gives the probability that the disease stage i lasts longer than s time units (or the probability of being still in the same stage at stage age s). Then, the derivative −P˙i (s) (i = E, I ) gives the rate of removal from the stage i at stage age s by the natural progression of the disease. These duration functions have the following properties: Pi (0) = 1,

P˙i (s) ≤ 0,



0

∞

Pi (s)ds < ∞,

i = E, I.

90

3 Endemic Disease Models

For the vital dynamics, we use the simplest function e−μt for the probability of survival (because our focus is on the effect of arbitrary distribution for disease stages). Let the numbers of initial susceptible and removed individuals be S0 > 0 and R0 > 0 respectively. Let E0 (t)e−μt and I0 (t)e−μt be the non-increasing functions that represent the numbers of individuals that were initially exposed and infective, respectively, and are still alive and in the respective classes at time t. E(0) and I (0) are constants representing the number of individuals in the E and I classes, respectively, at time t = 0. Let I˜0 (t) denote those initially infected who have moved into the I , and are still alive at time t. Consider the force of infection λ(t) that takes the form λ(t) = c

I (t) . N

(3.30)

Then the number of individuals who became exposed at some time s ∈ (0, t) and are still alive and in the E class at time t is  t λ(s)S(s)PE (t − s)e−μ(t−s) ds + E0 (t)e−μt , E(t) = 0

and the number of infectious individuals at time t is  t τ λ(s)S(s)[−P˙E (τ − s)]PI (t − τ )e−μ(t−s) dsdτ + I0 (t)e−μt + I˜0 (t). I (t) = 0

0

Assume that the recruitment rate is μN and they all enter the S class. Then the SEI R model reads  t  t −μ(t−s) μNe ds − λ(s)S(s)e−μ(t−s) ds + S0 e−μt , S(t) = 0 0 t (3.31) λ(s)S(s)PE (t − s)e−μ(t−s) ds + E0 (t)e−μt , E(t) =  0t  τ λ(s)S(s)[−P˙E (τ − s)]PI (t − τ )e−μ(t−s) dsdτ + I˜(t), I (t) = 0

0

˜ = X0 (t)e−μt + X˜ 0 (t) (X = Q, I, H, R). where λ(t) is given in (3.30), and X(t) It can be shown that under standard assumptions on initial data and parameter functions the system (3.31) has a unique non-negative solution defined for all positive time. When specific distributions are assumed for functions PE and PI the system (3.31) might be simplified. In particular, under the exponential distribution assumption (EDA) and gamma distribution assumption (GDA) the system can be reduced to be ODE systems, which will be referred to as the exponential distribution model (EDM) and gamma distribution model (GDM), respectively. This allows the examination of how the distribution assumptions may affect the model predictions.

3.6 *An SEI R Model with General Disease Stage Distributions

91

Let a(τ ) = e



−μτ

τ

0

[−P˙E (τ − u)]PI (u)du.

(3.32)



(3.33)

The reproduction number is given by Rc =

∞

ca(τ )dτ.

0

To see the biological meaning of the expression (3.33) and to simplify the notation in later sections we introduce the following quantities: TE = DE =



∞

0 ∞

[−P˙E (s)]e−μs ds, TI = PE (s)e−μs ds,

DI =

0



∞

0 ∞

[−P˙I (s)]e−μs ds, PI (s)e−μs ds.

(3.34)

0

TE , TI represent, respectively, the probability that exposed individuals survive and become infectious, and the probability that infectious individuals survive and become recovered. DE represents the mean sojourn time (death-adjusted) in the exposed stage, and DI represents the mean sojourn time (death-adjusted) in the infectious stage. Using (3.34) we can rewrite Rc in (3.33) as Rc = c



0

∞

a(τ )dτ = cTEk DIl .

(3.35)

System (3.31) always has the disease-free equilibrium (DFE), and an endemic equilibrium may exist depending on the value of R0 as described below. The proof of the result can be found in [18]. Result For System (3.31), the DFE is a global attractor if Rc < 1 and unstable if Rc > 1, in which case an endemic equilibrium exists and is stable. To compare the model behavior under different stage distributions, let PE and PI be the gamma distributions with the duration functions PE (s) = pm (s, κ) and PI (s) = pn (s, α), with mean 1/κ and 1/α, respectively. When m = n = 1, where m and n are the shape parameters, PE and PI are exponential distributions the general model (3.31) has the usual form of the standard SEIR model: S ′ = μN − cS NI − μS, E ′ = cS NI − (κ + μ)E, I ′ = κE − (α + μ)I.

(3.36)

For other integers m and n, then the integral equation model (3.31) can be reduced to an ordinary differential equation model. It has been noted that the use

92

3 Endemic Disease Models

of the gamma distribution pn (s, θ ) for a disease stage, e.g., the exposed stage, is equivalent to assuming that the entire stage is replaced by a series of n sub-stages, and each of the sub-stages is exponentially distributed with the removal rate nθ and the mean sojourn time T /n, where T = 1/θ is the mean sojourn time of the entire stage (see, for example, [23, 30, 32]). This approach of converting a gamma distribution to a sequence of exponential distributions is known as the “linear chain trick”. In this case, the general model (3.31) reduces to the following ODEs: S ′ = μN − cS NI − μS, E1′ = cS NI − (mκ + μ)E1 , Ej′ = mκEj −1 − (mκ + μ)Ej , j = 2, · · · , m, I1′ = mκEm − (nα + μ)I1 , Ij′ = nαIj −1 − (nα + μ)Ij , j = 2, · · · , n,

with I = nj=1 Ij .

(3.37)

From the formula (3.35) we get the reproduction number for system (3.37): Rc =

n−1 (mκ)m (nα)j c . (μ + mκ)m μ + nα (μ + nα)j

(3.38)

j =0

The qualitative behavior of the two systems (3.36) and (3.37) is the same due to the results stated above. For the quantitative behavior, some differences exist. For example, Fig. 3.7 illustrates that the model with gamma distributions tend to generate a lower frequency in oscillations than the model with exponential

Exponential Gamma

0.003

(a)

0.002

0.001

0

2

4

6 8 10 Time (years)

12

Fraction of infections

Fraction of infections

0.003

Exponential Gamma

(b)

0.002

0.001

0

2

4

6 8 10 Time (years)

12

Fig. 3.7 Comparison of simulations of the SEIR model with exponential and gamma distributions (3.36) and (3.37). Plot in (a) is for the case when the initial fraction of infectious individuals is higher (0.0015) while in (b) it is lower (0.0001). We observe in (a) that the solution of the model with gamma distribution has much lower magnitude in the oscillation than the solution of the model with exponential distribution, whereas in (b) it is opposite. However, in both (a) and (b) the frequency of the oscillation is lower for the gamma distribution model than for the exponential distribution model. The parameter values used are 1/κ = 2 (day), 1/α = 7 (day), 1/μ = 15 (year). This is more appropriate for modeling school entry and exit. The values of c is chosen so that R0 = 2

3.6 *An SEI R Model with General Disease Stage Distributions

93

distributions. As for the magnitude of oscillations, either model can have a higher magnitude than the other depending on the initial conditions.

3.6.1 *Incorporation of Quarantine and Isolation As pointed out earlier, the model with gamma distributions for disease stages (3.37) has similar qualitative behavior as the model with exponential distributions (3.36). As will be shown in this section, when control measures with quarantine and isolation are considered, models with exponential and gamma distributions can generate very different quantitative outcomes, including contradictory evaluations regarding control strategies. Let ρ denote the isolation efficiency with ρ = 1 representing complete effectiveness. Thus, when ρ < 1 the isolated individuals can transmit the infection with a reduced infectivity 1 − ρ. The force of infection is λ(t) = c

I (t) + (1 − ρ)H (t) . N

(3.39)

Let k(s), l(s): [0, ∞) → [0, 1] denote, respectively, the probabilities that exposed, infective individuals have not been quarantined, isolated at stage age s. ¯ ¯ give the respective probabilities of being Hence, 1 − k(s) =: k(s), 1 − l(s) =: l(s) quarantined, isolated before reaching stage age s. Assume that k(0) = l(0) = 1, ˙ ˙ k(s) ≤ 0 and l(s) ≤ 0. Consider the simpler case when the survivals from quarantine and isolation are described by the exponential functions k(s) = e−χ s ,

l(s) = e−φs

(3.40)

with χ and φ being constants, we have E0 (t) = E(0)e−(χ +α)t ,

I0 (t) = I (0)e−(φ+δ)t ,

etc.

(3.41)

The model with general distributions as well as quarantine and isolation is 

S(t) =

0



E(t) = Q(t) = I (t) =

t

μNe−μ(t−s) ds −

t 0

 t 0

0

t

λ(s)S(s)e−μ(t−s) ds + S0 e−μt ,

λ(s)S(s)PE (t − s)k(t − s)e−μ(t−s) ds + E0 (t)e−μt ,

 t 0



τ 0 τ

0

 

˙ −s)]PE t−τ τ −s e−μ(t−s) dsdτ +Q(t), ˜ λ(s)S(s)[−PE (τ −s)k(τ

λ(s)S(s)[−P˙E (τ −s)k(τ −s)]PI (t−τ )l(t − τ )e−μ(t−s) dsdτ +I˜(t),

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3 Endemic Disease Models

H (t) =

 t  u 0

0

τ 0

˙ − τ )] λ(s)S(s)[−P˙E (τ − s)k(τ − s)][−PI (u − τ )l(u  

×PI t − uu − τ e−μ(t−s) dsdτ du

 t τ ¯ − s)]PI (t − τ )e−μ(t−s) dsdτ +H˜ (t), λ(s)S(s)[−P˙E (τ − s)k(τ + 0

R(t) =

 t 0

0

0 τ

˜ λ(s)S(s)[−P˙E (τ − s)][1 − PI (t − τ )]e−μ(t−s) dsdτ + R(t), (3.42)

˜ = X0 (t)e−μt + X˜ 0 (t) (X = Q, I, H, R). where λ(t) is given in (3.39) and X(t) ˜ Again X(t) → 0 as t → ∞. Let  τ −μτ a1 (τ ) = e [−P˙E (τ − u)k(τ − u)]PI (u)l(u)du, 0 τ ¯ a2 (τ ) = e−μτ [−P˙E (τ − u)k(τ − u)]PI (u)l(u)du, (3.43) 0 τ ¯ − u)]PI (u)du, a3 (τ ) = e−μτ [−P˙E (τ − u)k(τ 0

¯ ¯ = 1 − l(s). Let where k(s) = 1 − k(s), l(s)



 A(τ ) = a1 (τ ) + (1 − ρ) a2 (τ ) + a3 (τ ) .

Then, the reproduction number is Rc = c



∞

A(τ )dτ.

(3.44)

0

We can also rewrite R0 in the following form: Rc = RI + RI H + RQH , where RI = c



∞

a1 (τ )dτ = cTEk DIl ,  ∞ RI H = (1 − ρ)c a2 (τ )dτ = (1 − ρ)cTEk (DI − DIl ), 0 ∞ RQH = (1 − ρ)c a3 (τ )dτ = (1 − ρ)c(TE − TEk )DI . 0

0

(3.45)

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95

and 

 ∞ −μs ˙ TE = [−PE (s)]e ds, TEk = [−P˙E (s)k(s)]e−μs ds,  0∞  0∞ TI = [−P˙I (s)]e−μs ds, TIl = [−P˙I (s)l(s)]e−μs ds, 0 ∞ 0 ∞ DE = PE (s)e−μs ds, DEk = PE (s)k(s)e−μs ds, 0 0  ∞  ∞ DI = PI (s)e−μs ds, DIl = PI (s)l(s)e−μs ds. ∞

0

(3.46)

0

The three components, RI , RI H , RQH in Rc represent contributions from the I class and from the H class through isolation and quarantine, respectively. TE and TEk represent, respectively, the probability and the “quarantine-adjusted” probability that exposed individuals survive and become infectious. TI and TIl represent, respectively, the probability and the “isolation-adjusted” probability that infectious individuals survive and become recovered. DE and DEk represent, respectively, the mean sojourn time (death-adjusted) and the “quarantine-adjusted” mean sojourn time (death-adjusted as well) in the exposed stage. DI and DIl represent, respectively, the mean sojourn time (death-adjusted) and the “isolationadjusted” mean sojourn time (death-adjusted as well) in the infectious stage. For system (3.42), the same results as for system (3.31) holds, i.e., the DFE is a global attractor if Rc < 1 and unstable if Rc > 1, in which case an endemic equilibrium exists and is stable.

3.6.2 *The Reduced Model of (3.42) Under GDA Again let PE and PI be the gamma distributions with the duration functions PE (s) = pm (s, κ) and PI (s) = pn (s, α), with mean 1/κ and 1/α, respectively. Then using the functions k(s) and l(s) given in (3.40) we can differentiate the equations in system (3.42) and obtain the following system of ordinary differential equations I + (1 − ρ)H − μS, N I + (1 − ρ)H E1′ = cS − (χ + mκ + μ)E1 , N

S ′ = μN − cS

Ej′ = mκEj −1 − (χ + mκ + μ)Ej , Q′1 Q′j

j = 2, · · · , m,

= χ E1 − (mκ + μ)Q1 , = χ Ej + mκQj −1 − (mκ + μ)Qj ,

I1′ = mκEm − (φ + nα + μ)I1 ,

j = 2, · · · , m, (3.47)

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3 Endemic Disease Models

Ij′ = nαIj −1 − (φ + nα + μ)Ij ,

j = 2, · · · , n,

H1′ = mκQm + φI1 − (nα + μ)H1 ,

Hj′ = nαHj −1 + φIj − (nα + μ)Hj ,

j = 2, · · · , n,

R ′ = nαIn + nαHn − μR, with I =

n j =1

Ij , H =

n

Hj .

j =1

In the special case when m = n = 1, the system (3.47) reduces to: S ′ = μN − cS I +(1−ρ)H − μS, N E ′ = cS I +(1−ρ)H − (χ + κ + μ)E, N Q′ = χ E − (κ + μ)Q, I ′ = κE − (φ + α + μ)I, H ′ = κQ + φI − (α + μ)H.

(3.48)

From the formula (3.45) we get the reproduction number for system (3.47): n−1 (mκ)m (nα)j c Rc = (μ + mκ)m μ + nδ (μ + nα)j



j =0

n−1 (nα)j  μ + nα (μ + mκ)m j =0 (μ+nα+φ)j 1−ρ 1− , (μ + mκ + χ )m μ + nα + φ n−1 (nα)j j 

j =0 (μ+nα)

(3.49)

with the derivatives

n−1 ∂Rc m(mκ)m (nα)j = −cρ < 0, m+1 ∂χ (μ + mκ + χ ) (μ + nα + φ)j +1

(3.50)

n−1 ∂Rc (mκ)m (j + 1)(nα)j = −cρ < 0. ∂φ (μ + mκ + χ )m (μ + nα + φ)j +2

(3.51)

j =0

j =0

3.6.3 *Comparison of EDM and GDM In this section, we show that when the GDA is used to replace the EDA, model predictions regarding the effectiveness of disease intervention policies may be different both quantitatively and qualitatively. We illustrate this by comparing the two models, GDM (3.47) and EDM (3.48). Two criteria are used in the comparison.

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97

One is the impact of control measures described by χ and φ on the reduction in the magnitude of Rc and the other one is the reduction in the number of cumulative infections C at the end of an epidemic (the final epidemic size). From (3.49) to (3.51) we know that the reproduction number Rc for GDM decreases with increasing χ and φ. Similarly, using the formula (3.45) we get the reproduction number for the EDM, Rc = RI + RI H + RQH , where RI = RI H

RQH

cκ , (μ + κ + χ )(μ + α + φ)

(1 − ρ)cκ = μ+κ +χ





 1 1 , − μ+α μ+α+φ

κ κ = (1 − ρ)c − μ+κ μ+κ +χ



1 , μ+α

which can be written in a simpler form as:    μ+κ c μ+α κ 1−ρ 1− . Rc = μ+κ μ+α μ+κ +χ μ+α+φ

(3.52)

The derivatives of Rc with respect to the control parameters are κ 1 ∂Rc = −cρ < 0, ∂χ (μ + κ + χ )2 μ + α + φ ∂Rc κ 1 = −cρ < 0. ∂φ μ + κ + χ (μ + α + φ)2 Hence, the reproduction number Rc for EDM also decreases as the control parameters χ and φ increase. Therefore, both models seem to work well when the impact of each individual control measure is considered. When we try to compare model predictions of combined control strategies, however, inconsistent predictions by the two models are observed. For example, in Fig. 3.8a, b, Rc for both models is plotted either as a function of φ for a fixed value of χ = 0.05, or as a function of χ for a fixed value φ = 0.05, or as a function of both χ and φ with χ = φ. For any vertical line except the one at 0.1, the three curves intersect the vertical line at three points that represent three control strategies. The order of these points (from top to bottom) determines the order of effectiveness (from low to high) of the corresponding control strategies since a larger Rc value will most likely lead to a higher disease prevalence. The order of these three points (labeled by a circle, a triangle, and a square) predicted by the EDM and the GDM is clearly different for the selected parameter sets, suggesting conflict assessments of interventions between the two models. These conflict assessments are also shown when we compare the C values. For example, Fig. 3.8a shows that the strategy corresponding to χ = 0.3, φ = 0.05 (indicated by the triangle) is more effective

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3 Endemic Disease Models

Fig. 3.8 Comparison of the EDM and the GDM on the impact of various control measures. (a) and (b) are plots of the reproduction number Rc as functions of control measures (χ and φ) for the EDM and GDM, respectively

than the strategy corresponding to χ = 0.05, φ = 0.3 (indicated by the solid circle). However, Fig. 3.8b shows the opposite, i.e., the strategy corresponding to χ = 0.3, φ = 0.05 (indicated by the triangle) is less effective than the strategy corresponding to χ = 0.05, φ = 0.3 (indicated by the solid circle). The parameter values used in Fig. 3.8 are c = 0.2, ρ = 0.8, κ = 1/7, and α = 1/10, corresponding to a disease with a latency period of 1/κ = 7 days and an infectious period of 1/α = 10 days (e.g., SARS). To examine in more detail the quantitative differences between the two models we conducted intensive simulations of the EDM and the GDM for various control measures, some of which are illustrated in Fig. 3.9. In this figure, the parameters for gamma distributions are m = n = 3, E and I represent the fraction of latent and infectious fractions E = (E1 + E2 + E3 )/N and I = (I1 + I2 + I3 )/N , respectively. The latent and infectious periods are κ = 1/7 and α = 1/10. The cumulative infection is calculated by integrating the incidence function, i.e., C(t) =



t 0

  cS(s) I (s) + (1 − ρ)H (s) /Nds

where H = H1 + H2 + H3 . Figure 3.9a, b is for Strategy I which implements quarantine alone with χ = 0.07, and Fig. 3.9c, d is for Strategy II which implements isolation alone with φ = 0.06. The effectiveness of these control measures is reflected by the corresponding C(t) values. According to Fig. 3.9a, c, the EDM predicts that Strategy II is more effective than Strategy I as the number C of cumulative infections (fractions) under Strategy II is 30% lower than the C value under Strategy I (notice that C ≈ 0.3 and C ≈ 0.2 under strategies I and II, respectively). However, according to Fig. 3.9b, d, the GDM predicts that Strategy

3.6 *An SEI R Model with General Disease Stage Distributions

99

Fig. 3.9 Comparison of control strategies and evaluations given by the exponential distribution model (EDM) and the gamma distribution model (GDM) with m = n = 3. Two strategies represented by χ and φ are compared: Strategy I involves quarantine alone (χ = 0.07 and φ = 0) while Strategy II involves isolation alone (χ = 0 and φ = 0.06). Other parameter values used are 1/κ = 7 (day), 1/α = 10 (day), 1/μ = 75 (year), and c = 0.2. The time unit is day

II is less effective than Strategy II as the number C of cumulative infections under Strategy I is 30% lower than the C value under Strategy II (notice that C ≈ 0.23 and C ≈ 0.36 under strategies I and II, respectively). Obviously, in this example, the predictions by the EDM and by the GDM are inconsistent. One of the main reasons for the discrepancy between models with exponential and gamma distributions is the memoryless property of the exponential distribution. This can be made more transparent by examining the expected remaining sojourns from the distributions. Under the gamma distribution pn (s, θ ) (or simply denoted by pn (s)) with n ≥ 2, the expected remaining sojourn at stage age s is Mn (s) =



∞ 0

1 pn (t + s) dt = pn (s) pn (s)



∞ s

1 pn (t)dt = nθ

n−1 k

(nθs)j j!

n−1 (nθs)k k=0 k!

k=0

j =0

.

After checking Mn′ (s) < 0 and lims→∞ Mn (s) → T /n where T = 1/θ , we know that Mn (s) strictly decreases with stage age s, and that when s is large the expected remaining sojourn can be as small as T /n. Hence, the expected remaining sojourn in a stage is indeed dependent on the time already spent in the stage. Therefore, the gamma distribution pn (s) for n ≥ 2 provides a more realistic description than the exponential distribution p1 (s) for which M1 (s) = T for all s.

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3 Endemic Disease Models

3.7 Diseases in Exponentially Growing Populations Many parts of the world experienced very rapid population growth in the eighteenth century. The population of Europe increased from 118 million in 1700 to 187 million in 1800. In the same time period the population of Great Britain increased from 5.8 to 9.15 million, and the population of China increased from 150 to 313 million [33]. The population of English colonies in North America grew much more rapidly than this, aided by substantial immigration from England, but the native population, which had been reduced to one tenth of their previous size by disease following the early encounters with Europeans and European diseases, grew even more rapidly. While some of these population increases may be explained by improvements in agriculture and food production, it appears that an even more important factor was the decrease in the death rate due to diseases. Disease death rates dropped sharply in the eighteenth century, partly from better understanding of the links between illness and sanitation and partly because the recurring invasions of bubonic plague subsided, perhaps due to reduced susceptibility. One plausible explanation for these population increases is that the bubonic plague invasions served to control the population size, and when this control was removed the population size increased rapidly. In developing countries it is quite common to have high birth rates and high disease death rates. In fact, when disease death rates are reduced by improvements in health care and sanitation it is common for birth rates to decline as well, since families no longer need to have as many children to ensure that enough children survive to take care of the older generations. Again, it is plausible to assume that population size would grow exponentially in the absence of disease but is controlled by disease mortality. The SI R model with births and deaths of Kermack and McKendrick [28] includes births in the susceptible class proportional to population size and a natural death rate in each class proportional to the size of the class. Let us analyze a model of this type with birth rate r and a natural death rate μ < r. For simplicity we assume the disease is fatal to all infectives with disease death rate α, so that there is no removed class and the total population size is N = S + I . Our model is S ′ = r(S + I ) − βSI − μS I ′ = βSI − (μ + α)I.

(3.53)

From the second equation we see that equilibria are given by either I = 0 or βS = μ + α. If I = 0, the first equilibrium equation is rS = μS, which implies S = 0 since r > μ. It is easy to see that the equilibrium (0,0) is unstable. What actually would happen if I = 0 is that the susceptible population would grow exponentially with exponent r − μ > 0. If βS = μ + α, the first equilibrium condition gives r

μ+α μ(μ + α) + rI − (μ + α)I − =0, β β

3.7 Diseases in Exponentially Growing Populations

101

which leads to (α + μ − r)I =

(r − μ)(μ + α) . β

Thus, there is an endemic equilibrium provided r < α + μ, and it is possible to show by linearizing about this equilibrium that it is asymptotically stable. On the other hand, if r > α + μ there is no positive equilibrium value for I . In this case we may add the two differential equations of the model to give N ′ = (r − μ)N − αI ≥ (r − μ)N − αN = (r − μ − α)N and from this we may deduce that N grows exponentially. For this model, either we have an asymptotically stable endemic equilibrium or population size grows exponentially. In the case of exponential population growth we may have either vanishing of the infection or an exponentially growing number of infectives. If only susceptibles contribute to the birth rate, as may be expected if the disease is sufficiently debilitating, the behavior of the model is quite different. Let us consider the model S ′ = rS − βSI − μS = S(r − μ − βI ) I ′ = βSI − (μ + α)I = I (βS − μ − α)

(3.54)

which has the same form as the Lotka–Volterra predator–prey model of population dynamics. This system has two equilibria, obtained by setting the right sides of each of the equations equal to zero, namely (0, 0) and an endemic equilibrium ((μ + α)/β, (r − μ)/β). It turns out that the qualitative analysis approach we have been using is not helpful as the equilibrium (0, 0) is unstable and the eigenvalues of the coefficient matrix at the endemic equilibrium have real part zero. In this case the behavior of the linearization does not necessarily carry over to the full system. However, we can obtain information about the behavior of the system by a method that begins with the elementary approach of separation of variables for first order differential equations. We begin by taking the quotient of the two differential equations and using the relation dI I′ = ′ S dS to obtain the separable first order differential equation dI I (βS − μ − α) = . dS S(r − βI ) Separation of variables gives       μ+α r β− dS . − β dI = I S

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3 Endemic Disease Models

Integration gives the relation β(S + I ) − r log I − (μ + α) log S = c where c is a constant of integration. This relation shows that the quantity V (S, I ) = β(S + I ) − r log I − (μ + α) log S is constant on each orbit (path of a solution in the (S, I ) plane). Each of these orbits is a closed curve corresponding to a periodic solution. We may view the model as describing an epidemic initially, leaving a susceptible population small enough that infection cannot establish itself. Then there is a steady population growth until the number of susceptibles is large enough for an epidemic to recur. During this growth stage the infective population is very small and random effects may wipe out the infection, but the immigration of a small number of infectives will eventually restart the process. As a result, we would expect recurrent epidemics. In fact, bubonic plague epidemics did recur in Europe for several hundred years. If we modify the demographic part of the model to assume limited population growth rather than exponential growth in the absence of disease, the effect would be to give behavior like that of the model studied in the previous section, with an endemic equilibrium that is approached slowly in an oscillatory manner if R0 > 1.

3.8 Project: Population Growth and Epidemics When one tries to fit epidemiological data over a long time interval to a model, it is necessary to include births and deaths in the population. Throughout the book we have considered population models with birth and death rates that are constant in time. However, population growth often may be fit better by assuming a linear population model with a time-dependent growth rate, even though this does not have a model-based interpretation. There could be many reasons for variations in birth and death rates; we could not quantify the variations even if we knew all of the reasons. Let r(t) = dN dt /N denote the time-dependent per capita growth rate. To estimate r(t) from linear interpolation of census data, proceed as follows: 1. Let Ni and Ni+1 be the consecutive census measurements of population size taken at times ti and ti+1 , respectively. Let ΔN = Ni+1 − Ni , Δt = ti+1 − ti , and δN = N (t + δt) − N(t). δN ΔN 2. If ti ≤ t ≤ ti+1 , ΔN Δt = δt , then we make the estimate r(t) ≈ ΔtN (t) . 3. A better approximation is obtained by replacing N (t) by N (t + δt/2). Why? (t)Δt −1 ) . Show that in this case, r(t) ≈ ( δt2 + NΔN

3.8 Project: Population Growth and Epidemics

103

Table 3.1 Population data growth for the USA Year 1700 1710 1720 1730 1740 1750 1760 1770 1780 1790 –

Population size 250,888 331,711 466,185 629,445 905,563 1,170,760 1,593,625 2,148,076 2,780,369 3,929,214 –

Year 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 –

Population size 5,308,483 7,239,881 9,638,453 12,866,020 17,069,453 23,192,876 31,443,321 39,818,449 50,155,783 62,947,714 –

Year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Population size 75,994,575 91,972,266 105,710,620 122,775,046 131,669,275 151,325,798 179,323,175 203,302,031 226,542,199 248,718,301 274,634,000

Fig. 3.10 Observed death rate (filled circle) and the best fit obtained with the function (3.55)

Question 1 Use the data of Table 3.1 to estimate the growth rate r(t) for the population of the USA. Figure 3.10 shows the time evolution of the USA mortality rate. This mortality rate is fit well by μ = μ0 +

μ0 − μf 1+e

′ )/Δ′ (t−t1/2

(3.55)

′ with μ0 = 0.01948, μf = 0.008771, t1/2 = 1912, and Δ′ = 16.61. Then the “effective birth rate” b(t) is defined as the real birth rate plus the immigration rate.

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3 Endemic Disease Models

Question 2 Estimate b(t) using r(t) = b(t) − μ(t), with r(t) found in Question 1. Consider an SEIR disease transmission model. We assume that: (a) An average infective individual produces β new infections per unit of time when all contacts are with susceptibles but that otherwise, this rate is reduced by the ratio S/N. (b) Individuals in the exposed class E progress to the infective class at the per capita rate k. (c) There is no disease-induced mortality or permanent immunity, and there is a mean infective period of 1/γ . We define γ = r + μ. The model becomes: dS I = bN − μS − βS , dt N dE I = βS − (k + μ)E, dt N dI = kE − (r + μ)I, dt dR = rI − μR. dt

(3.56)

Question 3 (a) Show that the mean number of secondary infections (belonging to the exposed class) produced by one infective individual in a population of susceptibles is Q0 = β/γ . (b) Assuming that k and μ are time-independent, show that R0 is given by Q0 f , where f = k/(k + μ). What is the epidemiological interpretation of Q0 f ? The usual measure of the severity of an epidemic is the incidence of infective cases. The incidence of infective cases is defined as the number of new infective individuals per year. If we take 1 year as the unit of time, the incidence of infective cases is given approximately by kE. The incidence rate of infective cases per 100,000 population is given approximately by 105 kE/N. Tuberculosis (TB) is an example of a disease with an exposed (noninfective) stage. Infective individuals are called active TB cases. Estimated incidence of active TB in the USA was in a growing phase until around 1900 and then experienced a subsequent decline. The incidence rate of active TB exhibited a declining trend from 1850 (see Table 3.2 and Fig. 3.11). The proportion of exposed individuals who k . The number f will be survive the latency period and become infective is f = k+μ used as a measure of the risk of developing active TB by exposed individuals. Question 4 Assume that the mortality rate varies according to the expression (3.55), and that the value of b found in Question 2 is used. Set γ = 1 years−1 and β = 10 years −1 , both constant through time. Simulate TB epidemics starting in 1700 assuming constant values for f . Can you reproduce the observed trends (Table 3.2)?

3.8 Project: Population Growth and Epidemics

105

Table 3.2 Reported incidence and incidence rate (per 100,000 population) of active TB Year 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1969 1970 1971 1972 1973 1974 1975

Incidence rate 53 49.3 46.9 41.6 39.2 36.5 32.5 30.8 29.4 28.7 28.7 26.6 25.3 24.4 23.1 19.4 18.3 17.1 15.8 14.8 14.2 15.9

Incidence 84,304 79,775 77,368 69,895 67,149 63,534 57,535 55,494 53,726 53,315 54,042 50,874 49,016 47,767 45,647 39,120 37,137 35,217 32,882 30,998 30,122 33,989

Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1992 1993 1994 1995 1996 1997 1998

Incidence rate 15 13.9 13.1 12.6 12.3 11.9 11 10.2 9.4 9.3 9.4 9.3 9.1 9.5 10.3 10.5 9.8 9.4 8.7 8 7.4 6.8

Incidence 32,105 30,145 28,521 27,769 27,749 27,337 25,520 23,846 22,255 22,201 22,768 22,517 22,436 23,495 25,701 26,673 25,287 24,361 22,860 21,337 19,885 18,361

It is not possible to obtain a good fit of the data of Table 3.2 to the model (3.56). It is necessary to use a refinement of the model that includes time-dependence in the parameters, and the next step is to describe such a model. The risk of progression to active TB depends strongly on the standard of living. An indirect measure of the standard of living can be obtained from the life expectancy at birth. The observed life expectancy for the USA is approximated well by the sigmoid shape function τ = τf +

(τ0 − τf ) , 1 + exp[(t − t1/2 )/Δ]

(3.57)

shown in Fig. 3.12. Here τ0 and τf are asymptotic values for life expectancy; t1/2 = 1921.3 is the time by which life expectancy reaches the value (τ0 + τf )/2; and Δ = 18.445 determines the width of the sigmoid. Assume that the risk f varies exactly like life expectancy, that is, assume that f is given by f (t) = ff +

(fi − ff ) . 1 + exp[(t − t1/2 )/Δ]

(3.58)

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3 Endemic Disease Models

Fig. 3.11 Incidence of active TB

Fig. 3.12 Observed average life expectancy at birth (filled circle) and its best fit (continuous line) using expression (3.57)

We refine the model (3.56) by replacing the parameter k by the variable expression μf (t)/(1 − f (t)) and k + μ by μ/(1 − f (t)), obtained from the relation f = k/(k + μ). Since the time scale of the disease is much faster than the demographic time scale, the recovery rate r is approximately equal to γ . This gives the model

3.9 *Project: An Environmentally Driven Infectious Disease

I dS = b(t)N − μ(t)S − βS , dt N dE I μ(t) = βS − E, dt N 1 − f (t) μ(t)f (t) dI = E − γ I, dt 1 − f (t) dR = γ I − μ(t)R. dt

107

(3.59)

Question 5 Simulate TB epidemics starting in 1700 using the model (3.59) with γ = 1 years−1 and β = 10 years−1 , both constant, and with μ(t) given by (3.55) and f (t) given by (3.58). Find values of f0 and ff for which an accurate reproduction of the observed TB trends (Table 3.2) is achieved. References: [1–4, 9–11, 15, 16, 38–41].

3.9 *Project: An Environmentally Driven Infectious Disease Consider an environmentally driven infectious disease such as cholera and toxoplasmosis (a parasite disease caused by T. gondii). For this type of disease, the transmission occurs when susceptible hosts have contacts with a contaminated environment, and the rate of environment contamination is dependent on both the number of infected hosts and the average pathogen load within an infected host. One way to model the transmission dynamics for such a disease is to consider both the disease transmission at the population level and the infection process within the hosts. The following model couples a simple within-host system for cell–parasite interactions (e.g., see [34–36]) and an endemic SI model with an interaction with a contaminated environment: T˙ = Λ − kV T − mT , T˙∗ = kV T − (m + d)T ∗ , V˙ = g(E) + pT ∗ − cV , S˙ = μ(S + I ) − λES − μS,

(3.60)

I˙ = λES − μI, E˙ = θ (V )I (1 − E) − γ E. Here, the variables for the within-host system T = T (t), T ∗ = T ∗ (t), and V = V (t) are the densities of healthy cells, infected cells, and parasite load, respectively. S = S(t) and I = I (t) denote the numbers of susceptible and infective individuals at time t, respectively. Λ denotes the recruitment rate of cells; k is the per-capita infection rate of cells; m and d are the per-capita background and infection-induced

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3 Endemic Disease Models

cell mortalities, respectively; p denotes the parasite production rate by an infected cell and c is the within-host clearance rate of pathogens. The variables S(t) and I (t) denote the numbers of susceptible and infective hosts at time t, and E(t) ( 0 ≤ E ≤ 1) represents the level of environmental contamination at time t, or the concentration of the pathogen per unit area of a region being considered. The parameter λ denotes the per-capita infection rate of hosts in a contaminated environment; μ denotes per-capita birth and natural death rate of hosts; and γ denotes the rate of pathogen clearance in the environment. The function g(E) in the V equation represents an added rate in the change of parasite load due to the continuous ingestion of parasites by the host from a contaminated environment, and is assumed to have the following properties: g(0) = 0,

g(E) ≥ 0,

g ′ (E) > 0,

g ′′ (E) ≤ 0.

(3.61)

One of the simplest forms for g(E) is the linear function g(E) = aE, where a is a positive constant. Other forms of g(E) include g1 (E) = aE/(1 + bE) with a and b being positive constants and g2 (E) = aE q (q < 1). For the analysis of the coupled model (3.60), a commonly used approach is to consider that the within-host system (consisting of the T , T ∗ , and V equations) occurs on a much faster time scale than the between-host system (consisting of S, I , and E equations), which allows the substitution of a stable equilibrium of the fastsystem (treating the slow-variables as constant) into the slow-system and study the lower-dimensional slow system (see, e.g., [7, 12, 17, 19]). The system for the fast variables is T˙ = Λ − kV T − mT T˙∗ = kV T − (m + d)T ∗ V˙ = g(E) + pT ∗ − cV ,

(3.62)

and the system for the slow variables is S˙ = μ(S + I ) − λES − μS, I˙ = λES − μI, E˙ = θ I V (1 − E) − γ E.

(3.63)

Question 1 Consider the fast system (3.62). The within-host reproduction number Rw (w for within) is given by Rw = where T0 = Λ/μ.

kpT0 c(m + d)

(3.64)

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109

(a) Let E > 0 be a constant. Show that (3.62) has a unique biologically feasible equilibrium (which depends on E) U˜ (E) = (T˜ (E), T˜ ∗ (E), V˜ (E)). (b) Show that the unique equilibrium U˜ (E) = (T˜ (E), T˜ ∗ (E), V˜ (E)) of (3.62) is globally, asymptotically stable. Hint: Consider the following Lyapunov function L (T , T ∗ , V ) = T˜

 ∗   T T T∗ T − log − 1 + T˜ ∗ − log −1 ˜ T˜ T T˜ ∗ T˜ ∗ V m+d ˜ V − log − 1 . + V p V˜ V˜



Consider the case when Rw > 1. It can be verified that V˜ (0) = limE→0 V˜ (E) > 0. Note that the total population of hosts N = S + I remains constant for all t > 0. Thus, the fast system (3.62) can be reduced to a two-dimensional system (by ignoring the S equations). Note also that U˜ is g.a.s. in the fast system. We can replace the fast variable V in (3.62) by V˜ (E) and study the following fast system I˙ = λE(N − I ) − μI, E˙ = θ I V˜ (E)(1 − E) − γ E.

(3.65)

The reproduction number for the between-host system, which is denoted by Rb (b for between) and defined as Rb =

θ V˜ (0) λN . μ γ

(3.66)

Therefore, Rb represents the number of secondary infections through the environment by one infected individual during the entire infectious period in a completely susceptible host population and environment. ˆ denote a biologically feasible equilibrium for (3.65). Show that Let Wˆ = (Iˆ, E) ˆ ˆ Iˆ = λEN/(λ E+μ) and Eˆ is a solution of the equation F (E) = G(E), 0 < E < 1, where   pm  1−E g(E) + T0 − T˜ (E) , F (E) = c m+d (3.67) μγ γE + . G(E) = θN θ λN

Equivalently, Eˆ is a zero of the function H (E) = F (E) − G(E) with 0 < E < 1.

(a) Let Wˆ 0 = (0, 0) denote the infection-free equilibrium of (3.65). Show that Wˆ 0 locally asymptotically stable when Rb < 1 and unstable when Rb > 1.

110 Fig. 3.13 Plot of the function H (E) for different Rb value between 0.75 and 2. A zero of H (E) in (0, 1) corresponds to positive equilibrium of the slow system (3.65)

3 Endemic Disease Models

1

0.5

0

-0.5

-1 0

0.2

0.4

0.6

0.8

1

(b) Show that it is possible for the equation H (E) = F (E) − G(E) = 0 to have 0, 1, or 2 solutions in (0, 1). Hint: Show first that H ′′ (E) > 0 for 0 < E < 1. (c) Figure 3.13 illustrates a numerical plot of the function H (E) for various Rb (by varying λ) values between 0.75 and 2. Other parameter values used are: Λ = 6 × 103 , k = 1.5 × 10−6 , m = 0.3, d = 0.2, a = 4 × 105 , c = 50, Rw0 = 1.09 (p = 908), N = 104 , μ = 4×10−4 , θ = 1×10−10 , and γ = 0.02. What do you observe? How does the number of solutions of H (E) = 0 depend on Rb ? (d) From the plot in part (c) we can observe that there exists a lower bound RbL ∈ (0, 1) such that for all Rb ∈ (RbL , 1), the equation H (E) = F (E) − G(E) = 0 has two solutions in (0, 1), which correspond to two positive equilibria Wˆ i = (Iˆi , Eˆ i ) (i = 1, 2) with Iˆ2 > Iˆ1 . In this case, prove analytically that Wˆ 2 is locally asymptotically stable and Wˆ 1 is unstable. (Hint: Check the sign of the eigenvalues of the Jacobian matrix at Wˆ ). (e) For the full system (3.60), conduct numerical simulations to confirm the results stated in parts (a)–(d), which are obtained by separating the fast and slow systems. (i) Reproduce the Fig. 3.14 (left) by plotting the fraction of infected I (t)/N vs. time with several sets of initial conditions. Use the same parameter values as in Part (c) except that p = 850 (corresponding to Rb = 0.37 < 1), a = 5 × 105 , λ = 5.5 × 10−4 , and γ = 0.015. (ii) Reproduce the phase portrait shown in Fig. 3.14 (right), which illustrates one fast variable (V ) and one slow variable (E). Use the same parameter values as in Part (c) except that p = 103 (corresponding to Rb > 1) and a = 4 × 104 .

3.10 *Project: A Two-Strain Model with Cross Immunity

111

Fig. 3.14 Left: Time plot of the full system (3.60) for Rb ∈ (RbL , 1), in which case there are two stable equilibria, one with the infection-free and one with positive infection level. Right: Phase portrait of the full system (3.60) for Rb > 1, in which case there is a unique stable equilibrium

3.10 *Project: A Two-Strain Model with Cross Immunity This project concerns a two-strain model with cross immunity. Divide the population into ten different classes: susceptibles (S), infected with strain i (Ii ), primary infection), isolated with strain i (Qi ), recovered from strain i (Ri , as a result of primary infection), infected with strain i (Vi , secondary infection), given that the population had recovered from strains j = i, and recovered from both strains (W ). Let A denote the population of non-isolated individuals and let βi S(IAi +Vi ) be the rate at which susceptibles become infected with strain i. That is, the ith (i = j ) incidence rate is assumed to be proportional to both the number of susceptibles i) and the available modified proportion of i-infectious individuals, (Ii +V A . Let σij denote a measure of the cross-immunity provided by a prior infection with strain i to exposure with strain j (i = j ). Consider the following model dS = dt dIi = dt dQi = dt dRi = dt dVi = dt dW = dt A=

2

(Ii + Vi ) − μS, A i=1 (Ii + Vi ) βi S − (μ + γi + δi )Ii , A δi Ii − (μ + αi )Qi , (Ij + Vj ) γi Ii + αi Qi − βj σij Ri − μRi , A (Ii + Vi ) βi σij Rj − (μ + γi )Vi , j = i A

2 i=1 γi Vi − μW,

S + W + 2i=1 (Ii + Vi + Ri ). Λ−

βi S

(3.68) j = i

112

3 Endemic Disease Models

The basic reproduction number for strain i is Ri =

βi , μ + γi + δi

i = 1, 2.

Assume that σ12 = σ21 = σ . The values of reproduction numbers Ri and the crossimmunity levels σ determine the existence and stability of equilibrium points of the system (3.68). Let Ei denote the boundary equilibria where only strain i is present (i = 1, 2). Question 1 Consider the case when changes in Ri are due to changes in βi . Let f (R1 ) and g(R2 ) be the two functions given by R1  δ2 1− 1 + σ (R1 − 1) 1 + μ+γ 2 

f (R1 ) =



(3.69)

,

(3.70)

μ(μ+α1 ) (μ+γ1 )(μ+α1 )+α1 δ1

and g(R2 ) =



1 + σ (R2 − 1) 1 +

R2  1−

δ1 μ+γ1

μ(μ+α2 ) (μ+γ2 )(μ+α1 )+α2 δ2

and let σ1∗ and σ2∗ be the critical values such that f ′ (R1 ) ≡

∂f (R1 , σ )   ∗ = 0, σ1 ∂R1

g ′ (R2 ) ≡

∂g(R2 , σ )   ∗ = 0. σ2 ∂R2

(3.71)

Determine the properties of f and g and sketch these functions in the R1 − R2 plane. Question 2 (a) Determine the region(s) in the R1 − R2 plane for the existence of E1 and E2 . (b) Determine the conditions for the stabilities of E1 and E2 .

3.11 Exercises 1. Consider the following SEIR model with disease-induced mortality: I dS = μN − βS − μS, dt N dE I = βS − (κ + μ)E, dt N

3.11 Exercises

113

dI = κE − (γ + μ + δ)I, dt dR = γ I − μR, dt N = S + E + I + R, where δ denotes the per capita rate of disease related death. (a) Compute the basic reproduction number R0 . (b) Does the system have an endemic equilibrium? If yes, find the condition in terms of R0 . (c) Show that the endemic equilibrium is locally asymptotically stable whenever it exists. (d) Reduce the system to a three-dimensional system by introducing fractions u = S/N , x = E/N, y = I /N, z = R/N. 2. Show that the endemic equilibrium of (3.3) is asymptotically stable if R0 > 1. 3. Consider a population in which a fraction p ∈ (0, 1) of newborns are successfully vaccinated and assume permanent immunity after infection and vaccination. Assume that infectious individuals are treated at a per capita rate r. Let Rc denote the control reproduction number such that the disease-free equilibrium is locally asymptotically stable when Rc < 1. Consider a disease for which β = 0.86, γ = 1/14 days −1 , μ = 1/75 years −1 . Use the following SIR model to calculate the threshold immunity level pc such that Rc < 1 for p > pc . dS dt dI dt dR dt N

(a) (b) (c) (d) (e)

I = μN(1 − p) − βS − μS, N I = βS − (γ + μ + r)I, N = μNp + (γ + r)I − μR, = S + I + R.

Find pc in the absent of treatment (i.e., r = 0). Find pc when r = 0.2. Plot pc as a function of r. Plot Rc as a function of p and r Plot several contour curves of Rc in the (p, r) plane including the curve for Rc = 1.

4. Consider the SIRS model (3.10). (a) Find the expression for the fraction I ∗ /N of the infected individuals at the endemic equilibrium. (b) Explore the dependence of I ∗ /N on the immunity loss (θ ), particularly in the two extreme cases when the immunity period is very short or very long.

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3 Endemic Disease Models

5.∗ Consider the SIR model with delay (3.12). (a) Find the endemic equilibrium. (b) Let β = 0.86 and α = 1/14. Determine the threshold value ωc such that the stability of the endemic equilibrium switches its stability. 6. Consider the vaccination model (3.21). (a) Verify that Rc < 1 whenever there is a backward bifurcation. (b) Show how to choose ϕ to make Rc < R0 , assuming that all parameters other than ϕ are kept fixed. (c) Is it possible to improve the vaccine (decrease σ ) enough to make Rc < R0 , assuming all parameters other than σ are kept fixed? 7.∗ Consider the model with a gamma distribution (3.47) and the exponential distribution (3.48). Compare the behavior of the two models under the scenarios specified below. Assume that all parameters have the same values as in Fig. 3.10 except the control parameters χ and ϕ. (a) χ = ϕ = 0. Do you observe any differences in the disease prevalence between the two models? Explain why or why not. (b) Compare the two models under two strategies. Strategy I: χ = 0.08 and ϕ = 0, Strategy II: χ = 0 and ϕ = 0.08. Do you observe any differences in the disease prevalence between the two models? Explain why or why not.

References 1. Aparicio, J.P., A. Capurro, and C. Castillo-Chavez (2000) Markers of disease evolution: the case of tuberculosis, J. Theor. Biol., 215: 227–238. 2. Aparicio J. P.,A.F. Capurro, and C. Castillo-Chavez (2000) On the fall and rise of tuberculosis. Department of Biometrics, Cornell University, Technical Report Series, BU-1477-M. 3. Aparicio J.P., A. Capurro, and C. Castillo-Chavez (2002) Frequency Dependent Risk of Infection and the Spread of Infectious Diseases. In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases : Models, Methods and Theory, Edited by Castillo-Chavez, C. with S. Blower, P. van den Driessche, D. Kirschner, and A.A. Yakubu, Springer-Verlag (2002), pp. 341–350. 4. Aparicio, J.P., A. Capurro, and C. Castillo-Chavez (2002) On the long-term dynamics and re-emergence of tuberculosis. In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, Edited by Castillo-Chavez, C. with S. Blower, P. van den Driessche, D. Kirschner, and A.A. Yakubu, Springer-Verlag, (2002) pp. 351–360. 5. Bellman, R.E. and K.L. Cooke (1963) Differential-Difference Equations, Academic Press, New York. 6. Blower, S.M. & A.R. Mclean (1994) Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco, Science 265: 1451–1454. 7. Boldin, B. and O. Diekmann (2008) Superinfections can induce evolutionarily stable coexistence of pathogens, J. Math. Biol. 56: 635–672. 8. Busenberg, S. and K.L. Cooke (1993) Vertically Transmitted Diseases: Models and Dynamics, Biomathematics 23, Springer-Verlag, Berlin-Heidelberg-New York.

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9. Castillo-Chavez, C., and Z. Feng (1997) To treat or not to treat; the case of tuberculosis, J. Math. Biol, 35: 629–656. 10. Castillo-Chavez, C., and Z. Feng (1998) Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosc. 151: 135–154. 11. Castillo-Chavez, C. and Z. Feng (1998) Mathematical models for the disease dynamics of tuberculosis, Advances in Mathematical Population Dynamics - Molecules, Cells, and Man (O. Arino, D. Axelrod, M. Kimmel, (eds)): 629–656, World Scientific Press, Singapore. 12. Cen, X., Z. Feng and Y. Zhao (2014) Emerging disease dynamics in a model coupling withinhost and between-host systems, J. Theor. Biol. 361: 141–151. 13. Conlan, A.J.K. and B.T. Grenfell (2007) Seasonality and the persistence and invasion of measles, Proc. Roy. Soc. B. 274: 1133–1141. 14. Dushoff, J., W. Huang, & C. Castillo-Chavez (1998) Backwards bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol. 36: 227–248. 15. Feng, Z., C. Castillo-Chavez, and A. Capurro (2000) A model for TB with exogenous reinfection, J. Theor. Biol. 5: 235–247. 16. Feng, Z., W. Huang and C. Castillo-Chavez (2001) On the role of variable latent periods in mathematical models for tuberculosis, J. Dynamics and Differential Equations 13: 425–452. 17. Feng, Z., J. Velasco-Hernandez and B. Tapia-Santos (2013) A mathematical model for coupling within-host and between-host dynamics in an environmentally-driven infectious disease, Math. Biosc. 241: 49–55. 18. Feng, Z., D. Xu, & H. Zhao (2007) Epidemiological models with non-exponentially distributed disease stages and applications to disease control, Bull. Math. biol. 69: 1511–1536. 19. Gilchrist, M.A. and D. Coombs (2006) Evolution of virulence: interdependence, constraints, and selection using nested models, Theor. Pop. Biol. 69: 145–153. 20. Hadeler, K.P. & C. Castillo-Chavez (1995) A core group model for disease transmission, Math Biosc. 128: 41–55. 21. Hadeler, K.P. and P. van den Driessche (1997) Backward bifurcation in epidemic control, Math. Biosc. 146: 15–35. 22. Hethcote, H.W. (1978) An immunization model for a heterogeneous population, Theor. Pop. Biol. 14: 338–349. 23. Hethcote, H.W. and D.W. Tudor (1980) Integral equation models for endemic infectious diseases, J. Math. Biol. 9: 37–47. 24. Hethcote, H.W., H.W. Stech, and P. van den Driessche (1981) Periodicity and stability in epidemic models: A survey, in Differential Equations and Applications in Ecology, Epidemics, and Population Problems, S. Busenberg and K.L. Cooke (eds.), Academic Press, New York, pages 65–82. 25. Hopf, E. (1942) Abzweigung einer periodischen Lösungen von einer stationaren Lösung eines Differentialsystems, Berlin Math-Phys. Sachsiche Akademie der Wissenschaften, Leipzig, 94: 1–22. 26. Hurwitz, A. (1895) Über die Bedingungen unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen bezizt, Math. Annalen 46: 273–284. 27. Keeling, M.J. and B.T. Grenfell (1997) Disease extinction and community size: Modeling the persistence of measles, Science 275: 65–67. 28. Kermack, W.O. and A.G. McKendrick (1932) Contributions to the mathematical theory of epidemics, part. II, Proc. Roy. Soc. London, 138: 55–83. 29. Kribs-Zaleta, C.M. and J.X. Velasco-Hernandez (2000) A simple vaccination model with multiple endemic states, Math Biosc. 164: 183–201. 30. Lloyd, A. L. (2001) Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods, Proc. Royal Soc. London. Series B: Biological Sciences 268: 985–993. 31. Lloyd, A. L. (2001b) Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and 60 (1), 59–71. 32. MacDonald, N. (1978) Time lags in biological models (Vol. 27), Heidelberg, Springer-Verlag. 33. McNeill, W.H. (1976) Plagues and Peoples, Doubleday, New York.

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34. Nowak, M.A., S. Bonhoeffer, G. M. Shaw and R. M. May (1997) Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations, J. Theor. Biol. 184: 203–217. 35. Perelson, A.S., D. E. Kirschner and R. De Boer (1993) Dynamics of HIV infection of CD4+ T cells, Math. Biosc. 114: 81–125. 36. Perelson, A.S. and P. W. Nelson (2002) Modelling viral and immune system dynamics, Nature Rev. Immunol. 2: 28–36. 37. Routh, E.J. (1877) A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion, MacMillan. 38. U. S. Bureau of the Census (1975) Historical statistics of the United States: colonial times to 1970, Washington, D. C. Government Printing Office. 39. U.S. Bureau of the Census (1980) Statistical Abstracts of the United States, 101st edition. 40. U.S. Bureau of the Census (1991) Statistical Abstracts of the United States, 111th edition. 41. U.S. Bureau of the Census (1999) Statistical Abstracts of the United States, 119th edition. 42. Wang, W. (2006) Backward bifurcations of an epidemic model with treatment, Math. Biosc. 201: 58–71. 43. Wang, W. and S. Ruan (2004) Bifurcations in an epidemic model with constant removal rate of the infectives, J. Math. Anal. & Appl. 291: 775–793. 44. Wearing, H. J., P. Rohani, & M. J. Keeling (2005) Appropriate models for the management of infectious diseases, PLoS Medicine 2 (7), e174.

Chapter 4

Epidemic Models

In this chapter we describe models for epidemics, acting on a sufficiently rapid time scale that demographic effects, such as births, natural deaths, immigration into and emigration out of a population may be ignored. The prototype epidemic model is the simple Kermack–McKendrick model studied in Sect. 2.4. We have established that the simple Kermack–McKendrick epidemic model S ′ = −βSI I ′ = βSI − αI

(4.1)

has the basic properties: 1. There is a basic reproduction number R0 such that if R0 < 1, the disease dies out while if R0 > 1 there is an epidemic. 2. The number of infectives always approaches zero and the number of susceptibles always approaches a positive limit S∞ as t → ∞. 3. There is a relation   S∞ S0 (4.2) log = R0 1 − S∞ N between the reproduction number and the final size of the epidemic which is an equality if there are no disease deaths. We will extend this model to models with more compartments and general distributions of stay in a compartment, for which these properties also hold, but we begin this chapter by presenting a more realistic description of the beginning of a disease outbreak than the compartmental approach. We will assume throughout this chapter that there are no disease deaths; the effects of including disease deaths in an epidemic model are the same as those described for the Kermack–McKendrick model with disease deaths in Sect. 2.5. © Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_4

117

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4 Epidemic Models

4.1 A Branching Process Disease Outbreak Model The Kermack–McKendrick compartmental epidemic model assumes that the sizes of the compartments are large enough that the mixing of members is homogeneous, or at least that there is homogeneous mixing in each subgroup if the population is stratified by activity levels. However, at the beginning of a disease outbreak, there is a very small number of infective individuals and the transmission of infection is a stochastic event depending on the pattern of contacts between members of the population; a description should take this pattern into account. Our approach will be to give a stochastic branching process description of the beginning of a disease outbreak to be applied so long as the number of infectives remains small, distinguishing a (minor) disease outbreak confined to this stage from a (major) epidemic, which occurs if the number of infectives begins to grow at an exponential rate. Once an epidemic has started we may switch to a deterministic compartmental model, arguing that in a major epidemic, contacts would tend to be more homogeneously distributed. Implicitly, we are thinking of an infinite population, and by a major epidemic, we mean a situation in which a nonzero fraction of the population is infected, and by a minor outbreak, we mean a situation in which the infected population may grow but remains a negligible fraction of the population. There is an important difference between the behavior of branching process models and the behavior of models of Kermack–McKendrick type, namely that, as we shall see in this section, for a stochastic disease outbreak model if R0 < 1 the probability that the infection will die out is 1, but if R0 > 1 there is a positive probability that the infection will increase initially but will produce only a minor outbreak and will die out before triggering a major epidemic. We describe the network of contacts between individuals by a graph with members of the population represented by vertices and with contacts between individuals represented by edges. The study of graphs originated with the abstract theory of Erdös and Rényi of the 1950s and 1960s [13–15]. It has become important in many areas of application, including social contacts and computer networks, as well as the spread of communicable diseases. We will think of networks as bidirectional, with disease transmission possible in either direction along an edge. An edge is a contact between vertices that can transmit infection. The number of edges of a graph at a vertex is called the degree of the vertex. The degree distribution of a graph is {pk }, where pk is the fraction of vertices having degree k. The degree distribution is fundamental in the description of the spread of disease. We think of a small number of infectives in a population of susceptibles large enough that in the initial stage we may neglect the decrease in the size of the susceptible population. Our development begins along the lines of that of [12] and then develops along the lines of [10, 32, 34]. We assume that the infectives make contacts independently of one another and let pk denote the probability that

∞ the number of contacts by a randomly chosen individual is exactly k, with k=0 pk = 1. In other words, {pk } is the degree distribution of the vertices of

4.1 A Branching Process Disease Outbreak Model

119

the graph corresponding to the population network. For the moment, we assume that every contact leads to an infection, but we will relax this assumption later. It is convenient to define the generating function G0 (z) =

∞

pk z k .

k=0

Since ∞ k=0 pk = 1, this power series converges for 0 ≤ z ≤ 1, and may be differentiated term by term. Thus (k)

pk =

G0 (0) , k!

k = 0, 1, 2, · · · .

It is easy to verify that the generating function has the properties G0 (0) = p0 ,

G0 (1) = 1,

G′0 (z) > 0,

G′′0 (z) > 0.

The mean degree, which we denote by k or z1 , is

k =

∞ k=1

kpk = G′0 (1).

More generally, we define the moments

k j  =

∞ k=1

k j pk ,

j = 1, 2, · · · ∞.

When a disease is introduced into a network, we think of it as starting at a vertex (patient zero) who transmits infection to every individual to whom this individual is connected, that is, along every edge of the graph from the vertex corresponding to this individual. We may think of this individual as being inside the population, as when a member of a population returns from travel after being infected, or as being outside the population, as when someone visits another population and brings back an infection. For transmission of disease after this initial contact, we need to use the excess degree of a vertex. If we follow an edge to a vertex, the excess degree of this vertex is one less than the degree. We use the excess degree because infection cannot be transmitted back along the edge whence it came. The probability of reaching a vertex of degree k, or excess degree (k − 1), by following a random edge is proportional to k, and thus the probability that a vertex at the end of a random edge has excess degree (k −1) is a constant multiple of kpk with the constant chosen to make the sum over k of the probabilities equal to 1. Then the probability that a

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4 Epidemic Models

vertex has excess degree (k − 1) is qk−1 =

kpk .

This leads to a generating function G1 (z) for the excess degree G1 (z) =

∞ k=1

qk−1 zk−1 =

∞ kpk k−1 1 z G′ (z), =

0 k=1

and the mean excess degree, which we denote by < ke >, is < ke > = = =

∞

1 k(k − 1)pk

k=1 ∞

∞

k=1

k=1

1 1 2 k pk − kpk

k2

< > − 1 = G′1 (1).

We let R0 = G′1 (1), the mean excess degree. This is the mean number of secondary cases caused by patient zero over the course of the outbreak and is the basic reproduction number as usually defined; the threshold for an epidemic is determined by R0 . The quantity < ke >= G′1 (1) is sometimes written in the form < ke >= G′1 (1) =

z2 , z1

2 where z2 = ∞ k=1 k(k − 1)pk =< k > − < k > is the mean number of second neighbors of a random vertex. We note that R0 depends not only on the mean degree but also on its variance. Our next goal is to calculate the probability that the infection will die out and will not develop into a major epidemic, proceeding in two steps. First we find the probability that a secondary infected vertex (a vertex which has been infected by another vertex in the population) will not spark a major epidemic. The computations are performed on a branching approximation to an actual epidemic valid only at the beginning of the epidemic. In an actual epidemic some contacts are “wasted” on individuals who have already been infected. Suppose that the secondary infected vertex has excess degree j . We let zn denote the probability that this infection dies out within the next n generations. For the infection to die out in n generations each of the j secondary infections coming from the initial secondary infected vertex must die out in (n − 1) generations. The probability of this is zn−1 for each secondary infection, and the probability that

4.1 A Branching Process Disease Outbreak Model

121 j

all secondary infections will die out in (n − 1) generations is zn−1 . Now zn is the sum over j of these probabilities, weighted by the probability qj of j secondary infections. Thus zn =

∞ j =0

j

qj zn−1 = G1 (zn−1 ),

z0 = 0.

We assume G1 (0) ≥ 0 so that there is an interval 0 < z ≤ w on which G1 (z) ≥ z; w is the smallest positive solution of z = G1 (z) and zn < w. If G1 (0) = 0, then q0 = 0, and p1 = 0. Thus the assumption G1 (0) > 0 is that the contact graph has no vertices with only one edge. The sequence zn is an increasing sequence and has a limit z∞ , which is the probability that this infection will die out eventually. Then z∞ is the limit as n → ∞ of the solution of the difference equation zn = G1 (zn−1 ),

z0 = 0.

Thus z∞ must be an equilibrium of this difference equation, that is, a solution of z = G1 (z). Since w is the smallest positive solution of z = G1 (z), z ≤ G1 (z) ≤ G1 (w) = w for 0 ≤ z ≤ w. It follows by induction that zn ≤ w, n = 0, 1, · · · ∞. From this we deduce that z∞ = w. The equation G1 (z) = z has a root z = 1 since G1 (1) = 1. Because the function G1 (z) − z has a positive second derivative, its derivative G′1 (z) − 1 is increasing and can have at most one zero. This implies that the equation G1 (z) = z has at most two roots in 0 ≤ z ≤ 1. If R0 < 1 the function G1 (z) − z has a negative first derivative G′1 (z) − 1 ≤ G′1 (1) − 1 = R0 − 1 < 0 and the equation G1 (z) = z has only one root, namely z = 1. On the other hand, if R0 > 1 the function G1 (z) − z is positive for z = 0 and negative near z = 1 since it is zero at z = 1 and its derivative is positive for z < 1 and z near 1. Thus in this case the equation G1 (z) = z has a second root z∞ < 1. This root z∞ is the probability that an infection transmitted along one of the edges at the initial secondary vertex will die out, and this probability is independent of the excess degree of the initial secondary vertex. It is also the probability that an infection originating outside the population, such as an infection brought into the population under study from outside, will die out.

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4 Epidemic Models

Next, we calculate the probability that an infection originating at a primary infected vertex, such as an infection introduced by a visitor from outside the population under study, will die out. The probability that the disease outbreak will die out eventually is the sum over k of the probabilities that the initial infection in a vertex of degree k will die out, weighted by the degree distribution {pk } of the original infection, and this is ∞ k=0

k = G0 (z∞ ). pk z∞

To summarize this analysis, we see that if R0 < 1, the probability that the infection will die out is 1. On the other hand, if R0 > 1, there is a solution z∞ < 1 of G1 (z) = z and there is a probability 1−G0 (z∞ ) > 0 that the infection will persist, and will lead to an epidemic. However, there is a positive probability G0 (z∞ ) that the infection will increase initially but will produce only a minor outbreak and will die out before triggering a major epidemic. This distinction between a minor outbreak and a major epidemic, and the result that if R0 > 1, there may be only a minor outbreak and not a major epidemic are aspects of stochastic models not reflected in deterministic models. If contacts between members of the population are random, corresponding to the assumption of mass action in the transmission of disease, then the probabilities pk are given by the Poisson distribution pk =

e−c ck k!

for some constant c [9, pp. 142–143]. The generating function for the Poisson distribution is ec(z−1) . Then G1 (z) = G0 (z), and R0 = c, so that G1 (z) = G0 (z) = eR0 (z−1) . The commonly observed situation that most infectives do not pass on infection but there are a few “super-spreading events” [35] corresponds to a probability distribution that is quite different from a Poisson distribution, and could give a quite different probability that an epidemic will occur. For example, if R0 = 2.5 the assumption of a Poisson distribution gives z∞ = 0.107 and G0 (z∞ ) = 0.107, so that the probability of an epidemic is 0.893. The assumption that nine out of ten infectives do not transmit infection while the tenth transmits 25 infections gives G0 (z) = (z25 + 9)/10,

G1 (z) = z24 ,

z∞ = 0,

G0 (z∞ ) = 0.9,

4.1 A Branching Process Disease Outbreak Model

123

from which we see that the probability of an epidemic is 0.1. Another example, possibly more realistic, is to assume that a fraction (1 − p) of the population follows a Poisson distribution with constant r while the remaining fraction p consists of super-spreaders each of whom makes L contacts. This would give a generating function G0 (z) = (1 − p)er(z−1) + pzL G1 (z) =

r(1 − p)er(z−1) + pLzL−1 , r(1 − p) + pL

and R0 =

r 2 (1 − p) + pL(L − 1) . r(1 − p) + pL

For example, if r = 2.2, L = 10, p = 0.01, numerical simulation gives R0 = 2.5,

z∞ = 0.146,

so that the probability of an epidemic is 0.849. These examples demonstrate that the probability of a major epidemic depends strongly on the nature of the contact network. Simulations suggest that for a given value of the basic reproduction number the Poisson distribution is the one with the maximum probability of a major epidemic. It has been observed that in many situations there is a small number of long range connections in the graph, allowing rapid spread of infection. There is a high degree of clustering (some vertices with many edges) and there are short path lengths. Such a situation may arise if a disease is spread to a distant location by an air traveler. This type of network is called a small world network. Long range connections in a network can increase the likelihood of an epidemic dramatically.

4.1.1 Transmissibility Contacts do not necessarily transmit infection. For each contact between individual of whom one has been infected and the other is susceptible, there is a probability that infection will actually be transmitted. This probability depends on such factors as the closeness of the contact, the infectivity of the member who has been infected, and the susceptibility of the susceptible member. We assume that there is a mean probability T , called the transmissibility, of transmission of infection. The transmissibility depends on the rate of contacts, the probability that a contact will transmit infection, the duration time of the infection, and the susceptibility. Until now we have assumed that all contacts transmit infection, that is, that T = 1.

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In this section, we will continue to assume that there is a network describing the contacts between members of the population whose degree distribution is given by the generating function G0 (z), but we will assume in addition that there is a mean transmissibility T . When disease begins in a network, it spreads to some of the vertices of the network. Edges that are infected during a disease outbreak are called occupied, and the size of the disease outbreak is the cluster of vertices connected to the initial vertex by a continuous chain of occupied edges. The probability that exactly m infections are transmitted by an infective vertex of degree k is   k T m (1 − T )k−m . m We define Γ0 (z, T ) be the generating function for the distribution of the number of occupied edges attached to a randomly chosen vertex, which is the same as the distribution of the infections transmitted by a randomly chosen individual for any (fixed) transmissibility T . Then Γ0 (z, T ) = = =

∞ ∞

m=0 ∞ k=0

∞ k=0

k=m

pk



   k T m (1 − T )(k−m) zm pk m

k   k (zT )m (1 − T )(k−m) m

m=0



(4.3)

pk [zT + (1 − T )]k = G0 (1 + (z − 1)T ).

In this calculation we have used the binomial theorem to see that k   k (zT )m (1 − T )(k−m) = [zT + (1 − T )]k . m

m=0

Note that Γ0 (0, T ) = G0 (1−T ),

Γ0 (1, T ) = G0 (1) = 1,

Γ0′ (z, T ) = T G′0 (1+(z−1)T ).

For secondary infections we need the generating function Γ1 (z, T ) for the distribution of occupied edges leaving a vertex reached by following a randomly chosen edge. This is obtained from the excess degree distribution in the same way, Γ1 (z, T ) = G1 (1 + (z − 1)T )

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125

and Γ1 (0, T ) = G1 (1−T ),

Γ1 (1, T ) = G1 (1) = 1,

Γ1′ (z, T ) = T G′1 (1+(z−1)T ).

The basic reproduction number is now R0 = Γ1′ (1, T ) = T G′1 (1). The calculation of the probability that the infection will die out and will not develop into a major epidemic follows the same lines as the argument for T = 1. The result is that if R0 = T G′1 (1) < 1, the probability that the infection will die out is 1. If R0 > 1 there is a solution z∞ (T ) < 1 of Γ1 (z, T ) = z, and a probability 1 − Γ0 (z∞ (T ), T ) > 0 that the infection will persist, and will lead to an epidemic. However, there is a positive probability Γ1 (z∞ (T ), T ) that the infection will increase initially but will produce only a minor outbreak and will die out before triggering a major epidemic. Another interpretation of the basic reproduction number is that there is a critical transmissibility Tc defined by Tc G′1 (1) = 1. In other words, the critical transmissibility is the transmissibility that makes the basic reproduction number equal to 1. If the mean transmissibility can be decreased below the critical transmissibility, then an epidemic can be prevented. The measures used to try to control an epidemic may include contact interventions, that is, measures affecting the network such as avoidance of public gatherings and rearrangement of the patterns of interaction between caregivers and patients in a hospital, and transmission interventions such as careful hand washing or face masks to decrease the probability that a contact will lead to disease transmission.

4.2 Network and Compartmental Epidemic Models Compartmental models for epidemics are not suitable for describing the beginning of a disease outbreak because they assume that all members of a population are equally likely to make contact with a very small number of infectives. Thus, as we have seen in the preceding section, stochastic branching process models are better descriptions of the beginning of an epidemic. They allow the possibility that even if a disease outbreak has a reproduction number greater than 1, it may be only a minor outbreak and may not develop into a major epidemic. One possible approach to a more realistic description of an epidemic would be to use a branching process model

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4 Epidemic Models

initially and then make a transition to a compartmental model when the epidemic has become established and there are enough infectives that mass action mixing in the population is a reasonable approximation. Another approach would be to continue to use a network model throughout the course of the epidemic. In this section we shall indicate how a compartmental approach and a network approach are related. The development is taken from [30, 31, 43]. We assume that there is a known static configuration model (CM) network in which the probability that a node u has degree ku is P (ku ). We let G0 (z) denote the probability generating function of the degree distribution, G0 (z) =

∞

pk z k ,

k=0

with mean degree k = G′0 (1). The per-edge contact rate from an infected node is assumed to be β, and it is assumed that infected nodes recover at a rate α. We use an edge-based compartmental model because the probability that a random neighbor is infected is not necessarily the same as the probability that a random individual is infected. We let S(t) denote the fraction of nodes that are susceptible at time t, I (t) the fraction of nodes that are infective at time t, and R(t) the fraction of nodes that are recovered at time t. It is easy to write an equation for R ′ , the rate at which infectives recover. If we know S(t), we can find I (t), because a decrease in S gives a corresponding increase in I . Since S(t) + I (t) + R(t) = 1, we need only find the probability that a randomly selected node is susceptible. We assume that the hazard of infection for a susceptible node u is proportional to the degree ku of the node. Each contact is represented by an edge of the network joining u to a neighboring node. We let ϕI denote the probability that this neighbor is infective. Then the per-edge hazard of infection is λE = βϕI . Assuming that edges are independent, u’s hazard of infection at time t is λu (t) = ku λE (t) = ku βϕI (t). Consider a randomly selected node u and let θ (t) be the probability that a random neighbor has not transmitted infection to u at time t. Then the probability that u is susceptible is θ ku . Averaging over all nodes, we see that the probability that a random node u is susceptible is S(t) =

∞ k=0

P (k)[θ (t)]k = G0 (θ (t)).

(4.4)

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127

We break θ into three parts, θ = ϕS + ϕI + ϕR , with ϕS the probability that a random neighbor v of u is susceptible, ϕI the probability that a random neighbor v of u is infective but has not transmitted infection to u, and ϕR the probability that a random neighbor v has recovered without transmitting infection to u. Then the probability that v has transmitted infection to u is 1 − θ . Since infected neighbors recover at rate α, the flux from ϕI to ϕR is αϕI . Thus ϕR′ = αϕI . It is easy to see from this that R ′ = αI.

(4.5)

Since edges from infected neighbors transmit infection at rate β, the flux from ϕI to (1 − θ ) is βϕI . Thus θ ′ = −βϕI .

(4.6)

To obtain ϕI′ we need the flux into and out of the ϕI compartment. The incoming flux from ϕS results from infection of the neighbor. The outgoing flux to ϕR corresponds to recovery of the neighbor without having transmitted infection, and the outgoing flux to (1−θ ) corresponds to transmission without recovery. The total outgoing flux is (α + β)ϕI . To determine the flux from ϕS to ϕI , we need the rate at which a neighbor changes from susceptible to infective. Consider a random neighbor v of u; the probability that v has degree k is kp(k)/ k. Since there are (k − 1) neighbors of v that could have infected v, the probability that v is susceptible is θ k−1 . Averaging over all k, we see that the probability that a random neighbor v of u is susceptible is ϕS =

∞ kp(k) k=0

k

θ k−1 =

G′0 (θ ) . G′0 (1)

(4.7)

To calculate ϕR , we note that the flux from ϕI to ϕR and the flux from ϕI to (1−θ ) are proportional with proportionality constant α/β. Since both ϕR and (1 − θ ) start at zero, ϕR =

α (1 − θ ). β

(4.8)

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4 Epidemic Models

Now, using (4.6)–(4.8), and ϕI = θ − ϕS − ϕR , we obtain θ ′ = −βϕI = −βθ + βϕS + βϕR = −βθ + β

G′0 (θ ) + α(1 − θ ). G′0 (1)

(4.9)

We now have a dynamic model consisting of Eqs. (4.4), (4.5), (4.9), and S + I + R = 1. We wish to show a relationship between this set of equations and the simple Kermack–McKendrick compartmental model (4.1). In order to accomplish this, we need only show under what conditions we would have S ′ = −βSI . Differentiating (4.4) and using (4.6), we obtain S ′ = G′0 (θ )θ ′ = −G′0 (θ )βϕI . Consider a large population with N members, each making C ≤ N − 1 contacts, so that S = θC,

G′0 (θ ) =

CS ′ S (θ ), θ

and S ′ = −βCS

ϕI . θ

We now let C → ∞ (which implies N → ∞) in such a way that βˆ = βC remains constant. Then S ′ = −βˆ

ϕI . θ

We will now show that ϕI ≈ 1, θ and this will yield the desired approximation ˆ S ′ = −βSI.

(4.10)

The probability that an edge to a randomly chosen node has not transmitted infection is θ (assuming that the given target node cannot transmit infection), and

4.3 More Complicated Epidemic Models

129

the probability that in addition it is connected to an infected node is ϕI . Because βˆ = βC is constant and therefore bounded as C grows, only a fraction no greater than a constant multiple of I /C of edges to the target node may have transmitted infection from a node that is still infected. For large values of C, ϕI is approximately I . Similarly, θ is approximately 1 as C → ∞. Thus ϕI /θ ≈ I as C → ∞. This gives the desired approximate equation for S. The result remains valid if all degrees are close to the average degree as the average degree grows. The edge-based compartmental modeling approach that we have used can be generalized in several ways. For example, heterogeneity of mixing can be included. In general, one would expect that early infections would be in individuals having more contacts, and thus that an epidemic would develop more rapidly than a mass action compartmental model would predict. When contact duration is significant, as would be the case in sexually transmitted diseases, an individual with a contact would play no further role in disease transmission until a new contact is made, and this can be incorporated in a network model. The network approach to disease modeling is a rapidly developing field of study, and there will undoubtedly be fundamental developments in our understanding of the modeling of disease transmission. Some useful references are [6, 27–29, 32– 34, 39] In the remainder of this chapter, we assume that we are in an epidemic situation following a disease outbreak that has been modeled initially by a branching process. Thus we return to the study of compartmental models. We will study models having more compartmental structure than the simple Kermack–McKendrick SI R epidemic model (4.1).

4.3 More Complicated Epidemic Models 4.3.1 Exposed Periods In many infectious diseases there is an exposed period after the transmission of infection from susceptibles to potentially infective members but before these potential infectives develop symptoms and can transmit infection. To incorporate an exposed period with mean exposed period 1/κ we add an exposed class E and use compartments S, E, I, R and total population size N = S + E + I + R to give a generalization of the epidemic model (4.1) S ′ = −βSI

E ′ = βSI − κE ′

I = κE − αI. A flow chart is shown in Fig. 4.1.

(4.11)

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4 Epidemic Models

Fig. 4.1 Flow chart for the SEIR model (4.11)

In this model there are two infected compartments, namely E and I with different infectivities. The parameter β represents the rate of effective contacts. The infectivity of a member of E is zero, and therefore contacts between members of S and members of E do not produce new infections. The analysis of this model is similar to the analysis of (4.1), but with I replaced by E + I . That is, instead of using the number of infectives as one of the variables we use the total number of infected members, whether or not they are capable of transmitting infection. For the model (4.11) it is no longer possible to distinguish whether there is an epidemic or not by determining whether the number of infectives grows or decreases initially. A more general characterization is given by whether the equilibrium with all members of the population susceptible is unstable (epidemic) or asymptotically stable (no epidemic). We will use a situation in which the disease-free equilibrium is unstable as our definition of an epidemic. For the model (4.11), the matrix of the linearization at the equilibrium S = N, E = 0, I = 0 is ⎡ ⎤ 0 0 −βN ⎣0 −κ βN ⎦ . 0 κ −α

The eigenvalues of this matrix are zero and the eigenvalues of the 2 × 2 matrix are 

 −κ βN . κ −α

The zero eigenvalue corresponds to movement along the line of equilibria. The other two eigenvalues have negative real part, corresponding to stability of the equilibrium with respect to solutions starting near the equilibrium but not on the line of equilibria, and failure of an epidemic to develop (since the trace of the matrix is negative) if and only if the determinant of the matrix is positive. The determinant is κ(α − βN ), and this is positive if and only if R0 < 1. If there is an epidemic, the initial exponential growth rate is the largest eigenvalue of the matrix, and this is the largest root of the quadratic characteristic equation λ2 + (α + κ)λ − κ(βN − α) = 0.

4.3 More Complicated Epidemic Models

131

Since the constant term in this equation is negative if R0 > 1, there is one negative and one positive root, and the positive root is λ=

−(α + κ) +

 (α − κ)2 + 4κβN . 2

This is the initial exponential growth rate; note that it is not the same as the initial exponential growth rate for the SI R model (4.1). The effect of an exposed period is to decrease the initial exponential growth rate.

4.3.2 A Treatment Model One form of treatment that is possible for some diseases is vaccination to protect against infection before the beginning of an epidemic. For example, this approach is commonly used for protection against annual influenza outbreaks. A simple way to model this would be to reduce the total population size by the fraction of the population protected against infection. In reality such inoculations are only partly effective, decreasing the rate of infection and also decreasing infectivity if a vaccinated person does become infected. This may be modeled by dividing the population into two groups with different model parameters which would require some assumptions about the mixing between the two groups. This is not difficult but we will not explore this direction until Chap. 5 on heterogeneous mixing. If there is a treatment for infection once a person has been infected, this may be modeled by supposing that a fraction γ per unit time of infectives is selected for treatment, and that treatment reduces infectivity by a fraction δ. Thus effective contacts between members of S and members of T would produce only δ new infections per contact. Suppose that the rate of removal from the treated class is η. This leads to the SI T R model, where T is the treatment class, given by S ′ = −βS[I + δT ]

I ′ = βS[I + δT ] − (α + γ )I

(4.12)

′

T = γ I − ηT . A flow chart is shown in Fig. 4.2. It is reasonable to assume that treatment does not slow recovery, so that η ≥ α. Also, since the total time in the infectious and treatment stages should not be greater than the mean infective period, we should expect that 1 1 1 ≥ + . α γ η

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4 Epidemic Models

I

S

T R

Fig. 4.2 Flow chart for the SI T R model (4.12)

It is not difficult to prove, much as was done for the model (4.1) that S∞ = lim S(t) > 0, t→∞

lim I (t) = lim T (t) = 0.

t→∞

t→∞

In order to calculate the basic reproduction number, we may argue that an infective in a totally susceptible population causes βN new infections in unit time, and the mean time spent in the infective compartment is 1/(α + γ ). In addition, a fraction γ /(α + γ ) of infectives are treated. While in the treatment stage the number of new infections caused in unit time is δβN, and the mean time in the treatment class is 1/η. Thus R0 is R0 =

γ δβN βN + . α+γ α+γ η

(4.13)

It is also possible to establish the final size relation (4.2) by means very similar to those used for the simple model (4.1). We integrate the first equation of (4.12) to obtain  ∞ S0 log = β[I (t) + δT (t)]dt. S∞ 0 Integration of the third equation of (4.12) gives γ



∞ 0

I (t)dt = η



∞ 0

T (t)dt.

4.3 More Complicated Epidemic Models

133

Integration of the sum of the first two equations of (4.12) gives N − S∞ = (α + γ )



∞

I (t)dt.

0

Combination of these three equations and (4.13) gives (4.2). For the model (4.12), the matrix of the linearization at the equilibrium S = N, I = 0, T = 0 is ⎡ ⎤ 0 −βN −δβN ⎣0 βN − (α + γ ) δβN ⎦ . 0 γ −η

The eigenvalues of this matrix are zero and the eigenvalues of the 2 × 2 matrix are   βN − (α + γ ) δβN . γ −η

The nonzero eigenvalues have negative real part, corresponding to stability of the equilibrium and failure of an epidemic to develop if and only if the determinant of the matrix is positive and the trace is negative. The determinant is −βN (η + δγ ) + η(α + γ ), which is positive if and only if R0 < 1, and this condition implies that the trace is negative. If there is an epidemic, the initial exponential growth rate is the largest eigenvalue of the matrix, and this is the largest root of the quadratic characteristic equation λ2 + [βN − (α + γ + η)]λ + βN (η + δγ ) − η(α + γ ) = 0. Since the constant term in this equation is negative if R0 > 1, there is one negative and one positive root, and the positive root is the initial exponential growth rate. Notice that this rate depends on the treatment rate.

4.3.3 An Influenza Model In some diseases, such as influenza, at the end of a stage individuals may proceed to one of two stages. There is a latent period after which a fraction p of latent individuals L proceeds to an infective stage I , while the remaining fraction (1 − p) proceeds to an asymptomatic stage A, with infectivity reduced by a factor δ and a

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4 Epidemic Models

I S

L

R A

Fig. 4.3 Flowchart for the influenza model (4.14)

different period 1/η. The influenza model of [3, 5] is S ′ = −βS[I + δA] L′ = βS[I + δA] − κL I ′ = pκL − αI A′ = (1 − p)κL − ηA

(4.14)

and R0 = βN



 p δ(1 − p) . + α η

A flow chart is shown in Fig. 4.3. The same approach used in earlier examples leads to the same final size relation (4.2). The model (4.14) is an example of a differential infectivity model. In such models, also used in the study of HIV/AIDS [23], individuals enter a specific group when they become infected and stay in that group over the course of the infection. Different groups may have different parameter values. For example, for influenza infective and asymptomatic members may have different infectivities and different periods of stay in the respective stages.

4.3.4 A Quarantine–Isolation Model For an outbreak of a new disease, where no vaccine is available, isolation of diagnosed infectives and quarantine of people who are suspected of having been infected (usually by tracing of contacts of diagnosed infectives) are the only control measures available. We formulate a model to describe the course of an epidemic,

4.3 More Complicated Epidemic Models

135

originally introduced for modeling the SARS epidemic of 2002–2003 [19], when control measures are begun under the assumptions: 1. Exposed members may be infective with infectivity reduced by a factor εE , 0 ≤ εE < 1. 2. Exposed members who are not isolated become infective at rate κE . 3. We introduce a class Q of quarantined members and a class J of isolated (hospitalized) members and exposed members are quarantined at a proportional rate γQ in unit time (in practice, a quarantine will also be applied to many susceptibles, but we ignore this in the model). Quarantine is not perfect, but reduces the contact rate by a factor εQ . The effect of this assumption is that some susceptibles make fewer contacts than the model assumes. 4. Infectives are diagnosed at a proportional rate γJ per unit time and isolated. Isolation is imperfect, and there may be transmission of disease by isolated members, with an infectivity factor of εJ . 5. Quarantined members are monitored and when they develop symptoms at rate κQ they are isolated immediately. 6. Infectives leave the infective class at rate αI and isolated members leave the isolated class at rate αJ . These assumptions lead to the SEQI J R model [19] S ′ = −βS[εE E + εE εQ Q + I + εJ J ]

E ′ = βS[εE E + εE εQ Q + I + εJ J ] − (κE + γQ )E

Q′ = γ Q E − κ J Q

(4.15)

′

I = κE E − (αI + γJ )I

J ′ = κQ Q + γJ I − αJ J. The model before control measures are begun is the special case γQ = γJ = κQ = αJ = 0, Q = J = 0 of (4.15). It is the same as (4.11). A flow chart is shown in Fig. 4.4. We define the control reproduction number Rc to be the number of secondary infections caused by a single infective in a population consisting only of susceptibles with the control measures in place. It is analogous to the basic reproduction number but instead of describing the very beginning of the disease outbreak it describes the beginning of the recognition of the epidemic. The basic reproduction number is the value of the control reproduction number with γQ = γJ = κQ = αJ = 0.

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4 Epidemic Models

S

E

I R

Q

H

Fig. 4.4 Flowchart for the SEQI J R model (4.15)

We have already calculated R0 for (4.11) and we may calculate Rc in the same way but using the full model with quarantined and isolated classes. We obtain Rc =

βN κE εQ βN εE γQ εJ βN κE γJ εJ βN γQ βN εE + + + + , D1 D1 D2 D1 κQ αJ D1 D2 αJ D1

where D1 = γQ + κE and D2 = γJ + αI . Each term of Rc has an epidemiological interpretation. The mean duration in E is 1/D1 with contact rate βεE , giving a contribution to Rc of βN εE /D1 . A fraction κE /D1 goes from E to I , with contact rate β and mean duration 1/D2 , giving a contribution of βN κE /D1 D2 . A fraction γQ /D1 goes from E to Q, with contact rate εE εQ β and mean duration 1/κQ , giving a contribution of εE εQ βN γQ /D1 κQ . A fraction κE γJ /D1 D2 goes from E to I to J , with a contact rate of εJ β and a mean duration of 1/αJ , giving a contribution of εJ βN κE γJ /αJ D1 D2 . Finally, a fraction γQ /D1 goes from E to Q to J with a contact rate of εJ β and a mean duration of 1/αJ giving a contribution of εJ βN γQ /D1 αJ . The sum of these individual contributions gives Rc . In the model (4.15) the parameters γQ and γJ are control parameters which may be chosen in the attempt to manage the epidemic. The parameters ǫQ and ǫJ depend on the strictness of the quarantine and isolation processes and are thus also control measures in a sense. The other parameters of the model are specific to the disease being studied. While they are not variable, their measurements are subject to experimental error. The linearization of (4.15) at the disease-free equilibrium (N, 0, 0, 0, 0) has matrix ⎡

⎤ ǫE βN − (κE + γQ ) εE εQ β βN εJ βN ⎢ −κQ 0 0 ⎥ γQ ⎢ ⎥. ⎣ 0 −(αI + γJ ) 0 ⎦ κE γJ −αJ 0 κQ

4.4 An SIR Model with a General Infectious Period Distribution

137

The corresponding characteristic equation is a fourth degree polynomial equation whose leading coefficient is 1 and whose constant term is a positive constant multiple of 1 − Rc , thus positive if Rc < 1 and negative if Rc > 1. If Rc > 1 there is a positive eigenvalue, corresponding to an initial exponential growth rate of solutions of (4.15). If Rc < 1 it is possible to show that all eigenvalues of the coefficient matrix have negative real part, and thus solutions of (4.15) die out exponentially [42]. In order to show that analogues of the relation (4.2) and S∞ > 0 derived for the model (4.1) are valid for the management model (4.15), we begin by integrating the equations for S + E, Q, I, J, of (4.15) with respect to t from t = 0 to t = ∞, using the initial conditions S(0) + E(0) = N (0) = N,

Q(0) = I (0) = J (0) = 0 .

We continue by integrating the equation for S and then an argument similar to the one used for (4.1) but technically more complicated may be used to show that S∞ > 0 for the treatment model (4.15) and also to establish the final size relation log

  S∞ S0 . = Rc 1 − S∞ N

Thus the asymptotic behavior of the management model (4.15) is the same as that of the simpler model (4.1). In the various compartmental models that we have studied, there are significant common features. This suggests that compartmental models can be put into a more general framework. In fact, this general framework is the age of infection epidemic model originally introduced by Kermack and McKendrick in [24]. We will explore this generalization in Sect. 4.5.

4.4 An SIR Model with a General Infectious Period Distribution In the simple model (4.1) studied in Sect. 2.4 we have assumed that the infective period is exponentially distributed . Now let us consider an SI R epidemic model in a population of constant size N with mass action incidence in which P (s) is the fraction of individuals who are still infective at time s after having become infected. The model is S ′ = −βS(t)I (t) 

I (t) = I0 (t) +

0

t

[−S ′ (t − s)]P (s) ds.

(4.16)

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4 Epidemic Models

Here, I0 (t) is the number of individuals infective initially at t = 0 who are still infective at time t. Then I0 (t) ≤ (N − S0 )P (t), because if all initial infectives were newly infected we would have equality in this relation, and if some initial infectives had been infected before the starting time t = 0, they would recover earlier. We assume that P (s) is a non-negative, non-increasing  ∞ function with P (0) = 1. We assume also that the mean infective period 0 P (s)ds is finite. Since a ∞ single infective causes βN new infections in unit time and 0 P (s)ds is the mean infective period, it is easy to calculate 

R0 = βN

∞

P (s) ds.

0

In this model it would be possible to assume that not all contacts by infectives with susceptibles lead to new infections. This could be achieved by incorporating a reduction factor δ in the equation for S and in the distribution P . Since S is a nonnegative decreasing function, it follows as for (4.1) that S(t) decreases to a limit S∞ as t → ∞, but we must  t proceed differently to show that I (t) → 0. This will follow if we can prove that 0 I (s) ds is bounded as t → ∞. We have 

t

0

I (s) ds =



0

t

I0 (s) ds +

≤ (N − S0 ) ≤ (N − S0 ) ≤N



t





t

 t 0

s

0

[−S ′ (s − u)]P (u)duds

P (s)ds +

0 t 0

P (s)ds +

 t



0

t u

[−S ′ (s − u)]dsP (u)du

t

0

[S0 − S(t − u)]P (s)ds

P (s)ds.

0

∞ t Since 0 P (s)ds is assumed to be finite, it follows that 0 I (s) ds is bounded, and thence that I (t) → 0. Now integration of the first equation in (4.16) from 0 to ∞ gives S0 log =β S∞ and this shows that S∞ > 0.



0

∞

I (s)ds < ∞,

4.5 The Age of Infection Epidemic Model

139

If all initially infected individuals are newly infected, so that I0 (t) = (N − S0 )P (t), integration of the second equation of (4.16) gives 

0

∞

I (s) ds =



0

∞



I0 (s) ds +

= (N − S0 ) = (N − S0 )





= (N − S∞ )

∞ 0 ∞ 0



0

∞ s 0

[−S ′ (s − u)]P (u)duds

P (u)du + P (u)du +

∞





0

∞ ∞ u

∞ 0

[−S ′ (s − u)]dsP (u)du

[S0 − S∞ ]P (u)du

P (u)du

0

  S∞ , = R0 1 − N and this is the final size relation, identical to (4.2). If there are individuals who were infected before time t = 0, a positive term (N − S0 )



∞ 0

P (t)dt −



∞

I0 (t)dt

0

must be subtracted from the right side of this equation. The generalization to arbitrary infective periods in this section is a component of the age of infection epidemic model of [24], which also incorporates general compartmental structures. The examples of this section and the previous section are all special cases of the age of infection model.

4.5 The Age of Infection Epidemic Model The general epidemic model described by Kermack and McKendrick [24] included a dependence of infectivity on the time since becoming infected (age of infection). We let S(t) denote the number of susceptibles at time t and let ϕ(t) be the total infectivity at time t, defined as the sum of products of the number of infected members with each infection age and the mean infectivity for that infection age. We assume that on average, members of the population make a constant number βN of contacts in unit time. We let B(s) be the fraction of infected members remaining infected at infection age s and let π(s) with 0 ≤ π(s) ≤ 1 be the mean infectivity (if infected) at infection age s. Then we let A(s) = π(s)B(s),

140

4 Epidemic Models

the mean infectivity of members of the population with infection age s including those who are no longer infectious. Here, the factor π must include the probability that a contact will produce a new infection. We assume that there are no disease deaths, so that the total population size is a constant N. Since −S ′ (t − s) is the number of new infections at time (t − s) and A(s) is the remaining infectivity of these new infections at time t (infection age (s)), the total infectivity at time t is ϕ(t) =



∞ 0

[−S ′ (t − s)]A(s)ds.

It is assumed that the disease outbreak begins at time t = 0, so that S(u) = N for u < 0. There may be a discontinuity in S(u) at u = 0 corresponding to an initial infective distribution. We may write the age of infection epidemic model as S ′ = −βSϕ  ∞  [−S ′ (t − s)]A(s)ds = ϕ(t) = 0

(4.17) ∞

0

βS(t − s)ϕ(t − s)A(s)ds.

The parameter β continues to represent the rate of effective contacts, and the transmission probability is contained in ϕ. The basic reproduction number is R0 = βN



∞

A(s)ds.

(4.18)

0

We may write the model in a single equation S ′ (t) = βS(t)



∞ 0

A(s)S ′ (t − s)ds.

There is an invasion criterion. Initially, in a fully susceptible population when S(t) is close to S0 = N, we may replace S(t) by its initial value N in this single equation, giving a linear equation. The condition that this linear equation have a solution S(t) = Nert is 1 = βN



∞

A(s)e−rs ds.

(4.19)

0

The result that the initial exponential growth rate in an epidemic is given by the solution of (4.19) was given in [20, 36] and later in [44]. It allows us to estimate the basic reproduction number for an epidemic, provided we know not only the mean infective period but also the distribution of infective periods. This result is valid for age of infection epidemic models, and thus is applicable to compartmental models

4.5 The Age of Infection Epidemic Model

141

that can be interpreted in an age of infection context, that is, provided we are able to calculate the infectivity function A(s). Combination of the relations (4.18) and (4.19) gives a relation between the initial exponential growth rate r and the basic reproduction number R0 , namely ∞

A(s)ds . R0 =  ∞0 −rs A(s)ds 0 e

Then R0 > 1 if and only if r > 0. This relation provides a means to estimate the basic reproduction number from measurements of the initial exponential growth rate provided the infectivity distribution is known. We define an epidemic as a situation in which for the model we have r > 0, so that initially the solution grows exponentially. Division of the model equation (4.17) by S(t) and integration with respect to t from 0 to ∞ gives, with an interchange of order of integration gives S0 = log S∞



∞

0

[S(−s) − S∞ ]A(s)ds.

Since we are assuming S(−s) = N if s > 0, we have log

S0 = S∞



0

∞

[N − S∞ ]A(s)ds,

and then we obtain the final size relation log

  S∞ S0 . = R0 1 − S∞ N

The final size relation is sometimes presented in the form log

  S0 S∞ , = R0 1 − S∞ S0

(4.20)

see for example [4, 12, 22], since normally S(0) ≈ N. This form would also represent the final size relation for an epidemic started by someone outside the population under study, so that S0 = N, I0 = 0.

4.5.1 A General SEIR Model We consider an SEIR model with general distributions of stay in both the exposed and infective period. The model is related to the model (3.31), which also includes births and natural deaths. Suppose the fraction of exposed individuals who are still

142

4 Epidemic Models

in the exposed class s time units after being exposed is PE (s) and the fraction of individuals who are still in the infectious class s time units after entering the infectious class is PI (s), with PE (s), PI (s) non-negative, non-increasing functions such that  ∞  ∞ PE (0) = PI (0) = 1, PE (s)ds < ∞, PI (s)ds < ∞. 0

0

Then PE and PI represent survival probabilities in the classes E and I , respectively. The probability density function for E, which will appear in the model, is q(τ ) = −PE′ (τ ). We assume that E0 newly exposed members enter the exposed class at time t = 0. Then S ′ = −βSI E(t) = E0 PE (t) +



t 0

[−S ′ (u)]PE (t − u)du.

In this equation we are assuming that all effective contacts transmit infection. Differentiation of the equation for E(t) gives E ′ (t) = E0 PE′ (t) − S ′ (t) +



t 0

[−S ′ (u)]PE′ (t − u)du,

and this shows that the input to the infectious stage at time t is −E0 PE′ (t) −



t 0

[−S ′ (u)]PE′ (t − u)du.

If we assume that S ′ (u) = 0 for u < 0 and that S ′ (u) has a jump of −E0 at u = 0, then we may write the equation for E(t) as E(t) =



∞ 0

[−S ′ (s)]PE (t − s)ds.

From the equation for E(t), we see that the output from E to I is −E0 PE′ (t) −



0

t

[−S ′ (s)]PE′ (t − s)ds = −



∞ 0

[−S ′ (s)]PE′ (t − s)ds.

4.5 The Age of Infection Epidemic Model

143

Then I (t) = − −





0

0

∞ ∞ 0

S ′ (s)PE′ (t − s − u)PI (u)duds

0

[−S ′ (s)]PE′ (t − s − u)PI (u)duS ′ (s)ds.

∞ ∞

We now have 

I (t) =

∞

0

[−S ′ (s)]AI (t − s)ds,

with 

AI (z) = −

∞ 0

PE′ (z − v)PI (v)dv.

Then the model is S ′ = −βSI  E(t) = I (t) =

∞

 0∞ 0

[−S ′ (s)]PE (t − s)ds

(4.21)

′

[−S (s)]AI (t − s)ds,

which is in age of infection form with Φ = I and A(z) = AI (z). Then R0 = βN



∞

A(z)dz

0 ∞  z

= −βN  = βN

0 ∞

0

PE′ (z − u)PI (u)dudz

(4.22)

PI (u)du,

0

∞ using − 0 PE′ (v)dv = PE (0) − PE (∞) = 1. The initial exponential growth rate of the general SEIR model (4.21) satisfies βN



0

∞

e

−rs



0

s

[−PE′ (s − u)]PI (u)duds = 1,

144

4 Epidemic Models

which reduces to  ∞ q(v)e−rv dv e−ru PI (u)du 0 0  ∞  ∞ −rv = βN 1 − r e PE (v)dv e−ru PI (u)du,

1 = βN



∞

0

(4.23)

0

with the aid of integration by parts. It is important to remember that in this example we are assuming that effective contacts transmit new infections. We have not allowed for reductions in infectivity or susceptibility, but it would be possible to include these factors into the model. From (4.23) we see that the effect of an exposed period is to decrease the initial exponential growth rate. It also indicates that the result of using the relation (4.19) depends on knowledge of the compartmental structure of the model.

4.5.2 A General Treatment Model Consider the treatment model (4.12) of Sect. 4.3. We now extend this to an age of infection model with general infective and treatment staged distributions. Assume that the distribution of infective periods is given by PI (τ ), and the distribution of periods in treatment is given by PT (τ ). Then the SI T R model becomes S ′ (t) = −βS(t)[I (t) + δT (t)]  t I (t) = I0 PI (t) + [−S ′ (t − σ )]e−γ σ PI (σ ) dσ T (t) =



(4.24)

0

∞

0

γ I (t − σ )PT (σ ) dσ,

and ϕ(t) = I (t) + δT (t). Assuming that S(u) has a jump of −I0 at u = 0, we may write the equation for I as  ∞ I (t) = − [−S ′ (t − s)]e−γ s PI (s)ds. 0

Substituting the expression for I into the equation for T , we obtain T (t) = =



∞

γ

0



0

∞

γ





∞ 0 ∞ 0

[−S ′ (t − s − σ )]e−γ s PI (s)PT (σ )dsdσ [−S ′ (t − s − σ )]PT (σ )dσ e−γ s PI (s)ds

4.5 The Age of Infection Epidemic Model

= = =



∞

γ

0



∞

0



∞

0



∞

145

[−S ′ (t − v)]PT (v − s)dve−γ s PI (s)ds

s

[−S ′ (t − v)]γ



v 0

PT (v − s)e−γ s PI (s)dsdv

[−S ′ (t − v)]A(v)dv,

with A(v) =



0

v

γ PT (v − s)e−γ s PI (s)ds.

Writing ϕ(t) = I (t) + δT (t), we now have the model (4.24) in age of infection form (without transmission probabilities), S ′ (t) = −βS(t)ϕ(t)  ∞

ϕ(t) = −

0

From this we see that  R0 = βN

0

= βN



  [−S ′ (t − s)] e−γ s PI (s) + δA(s) ds.

∞ ∞

 e−γ s PI (s) + δA(s) ds

e

−γ s

0

PI (s)ds + βN δγ



0

∞ s 0

(4.25)

PT (s − u)duds].

With exponentially distributed infective and treatment periods, PI (s) e−αs , PT (s) = e−ηs we calculate R0 , obtaining R0 = βN



0

∞

  e−(α+γ )τ dτ 1 + δγ

  δγ βN 1+ , = α+γ η

∞ 0

e−ητ dτ

=



the same result as (4.13). An arbitrary choice of treatment period distribution with mean 1/η does not affect the quantity R0 , but different infective period distributions may have a significant effect. For example, let us take γ = 1 and assume the mean infective

146

4 Epidemic Models

period is 1. Then, with an exponential distribution, PI (τ ) = e−τ , 

0

∞

e−τ P (τ ) dτ =



0

∞

e−2τ dτ =

1 . 2

With an infective period of fixed length 1, 

0

∞

e−τ PI (τ ) dτ =



0

1

e−τ dτ = (1 − e−1 ) = 0.632.

Thus a model with an infective period of fixed length would lead to a basic reproduction number more than 25% higher than a model with an exponentially distributed infective period that has the same mean.

4.5.3 A General Quarantine/Isolation Epidemic Model To cope with a disease outbreak for which there is no (as yet) known treatment, the only methods available are isolation of diagnosed infective and quarantine of suspected exposed members of the population. This approach was used during the SARS epidemic of 2003, and modeled in [19]. An SEIR model with general exposed and infective periods as well as quarantine and isolation has been analyzed in [47]. While the model in [47] was deterministic, the analysis was from a probabilistic point of view. Here we add quarantine and isolation to the model (4.21) and derive some of the results of [47] from a compartmental approach. The model is related to the model (3.42), which includes births and natural deaths, and is more general because it does not assume that quarantine and isolation are perfectly effective. We move members from the exposed class to a quarantine class Q at rate ψ and from the infective class to an isolated (hospitalized) class H at rate ϕ. For simplicity, we assume that both quarantine and isolation are perfectly effective, so that no infections are transmitted from either quarantined or isolated members. Then we need not include Q or H in the model unless we wish to track the number of individuals quarantined and isolated. We need only adjust the model (4.21) to include the removals. With quarantine but not yet isolation, the new equation for E is  t E(t) = E0 e−ψt PE (t) + [−S ′ (s)]e−ψ(t−s) PE (t − s)ds. 0

The input to I at time u becomes E0 qE (u) +



0

u

[−S ′ (τ )]e−ψ(u−τ ) qE (u − τ )dτ.

4.5 The Age of Infection Epidemic Model

147

Now, I (t) is given by I (t) = E0



t

0

+E0



qE (u)e−ψ(t−u) PI (t − u)du t 0

[−S ′ (s)]qE (u − s)E −ψ(u−s) dsPI (t − u)du.

The first term in this expression may be written as I0 (t), and the second term may be simplified, using interchange of the order of integration in the iterated integral much as in the previous section, to yield 

t

0

[−S ′ (s)]T (t − s)ds

with T (v) =



v

0

qE (y)e−ψy PI (v − y)dy.

Then the model is S ′ = −βSI E(t) = E0 e

−ψt

I (t) = I0 (t) +

PE (t) + 

t 0



t 0

[−S ′ (s)]e−ψ(t−s) PE (t − s)ds

[−S ′ (s)]T (t − s)ds.

If we add isolation at a rate ϕ of infectives, we obtain I (t) = e−ϕt I0 (t) +



t 0

[−S ′ (s)]e−ϕ(t−s) T (t − s)ds,

and the quarantine/isolation model is S ′ = −βSI

 t E(t) = E0 e−ψt PE (t) + [−S ′ (s)]e−ψ(t−s) PE (t − s)ds 0  t [−S ′ (s)]e−ϕ(t−s) T (t − s)ds. I (t) = E0 e−ϕt I0 (t) + 0

(4.26)

148

4 Epidemic Models

The reproduction number is now a control reproduction number Rc (ψ, ϕ), depending on the control parameters ψ and ϕ, Rc (ψ, ϕ) = βN = βN = βN = βN









∞

e−ϕτ T (τ )dτ

0 ∞

e

−ϕτ

E0

0 ∞

τ 0

e−ψy q(y)

0 ∞



q(y)e−ψy PI (τ − y)dydτ



e−ψy q(y)dy

0

∞ y



 e−ϕ(τ −y) PI (τ − y)dτ dy

∞

e−ϕu PI (u)du.

0

If we know the functions PI and q, we may calculate the sensitivity of R0 to the quarantine and isolation rates and thus compare quarantine and isolation as management strategies. In [47], this expression is written in terms of the Laplace transforms of q and PI , Rc (ψ, ϕ) = βN E0 Lq (ψ)LPI (ϕ). By comparison of the mean infectivity functions of the models (4.21) and (4.26), we see that the effect of quarantine and/or isolation on an SEIR model is to decrease the initial exponential growth rate. As suggested by these examples, there are general methods for calculation of integrals involving A(τ ) without the necessity of calculating the function A explicitly [7, 48]. A naive view of epidemic models suggests that if we know the basic reproduction number, we can determine the size of an epidemic. There are several quite different problems with this view. We have seen that estimation of the basic reproduction number from early observed data depends on the structure of the model being assumed. For example, if there is an exposed period, the initial exponential growth rate is less than if infected individuals become infectious immediately but this does not affect the basic reproduction number. This implies that an estimate of the basic reproduction number from the initial exponential growth rate will be too small. A second problem is that in an epidemic, early data may be incomplete and inaccurate, and its use may lead to a poor estimate of the reproduction number.

4.6 The Gamma Distribution There is ample evidence that exponential distributions of stay in compartments are much less realistic than gamma distributions [16, 17, 26, 46]. This has already been suggested in Sect. 3.6. A gamma distribution P (τ ) with parameter n and period 1/α

4.6 The Gamma Distribution

149

can be represented as a sequence of n exponential distributions Pi (τ ) with period 1/nα. Thus an SI R epidemic model with a gamma distribution for the infectious period may be represented by the system S ′ = −βSI I1′ = βSI − nαI1 Ij′ = nαIj −1 − nαIj ,

(4.27) j = 2, 3, · · · , n.

We may view this as an age of infection model S ′ = −βSI I (t) = I0 (t) +



t 0

[−S ′ (t − τ )]P (τ )dτ,

(4.28)

in which P (τ ) represents the fraction of infectives with infection age τ . To use the age of infection interpretation, we need to determine the kernel P (τ ) in order to calculate its integral. We let uk (τ ) be the fraction of infected members with infection age τ in the k-th subinterval. Then u′1 = −nαu1 , u1 (0) = 1 u′j = nαuj −1 − nαuj , uj (0) = 0,

j = 2, 3, · · · , n.

(4.29)

It is easy to solve this system of differential equations recursively, and we obtain (nα)k−1 k−1 −nατ τ e , (k − 1)!

uk (τ ) =

k = 1 · · · n.

This gives the desired kernel P (τ ) =

n k=1

uk (τ ) =

In order to evaluate the integral 

t

k−1 −ct

e

∞ 0

n (nα)(k−1) k=1

(k − 1)!

τ (k−1) e−nατ .

(4.30)

P (τ )dτ , we start with the integration formula

1 k−1 dt = − t k−1 e−ct + c c



t k−2 e−ct dt,

(4.31)

obtained using integration by parts; then with limits of integration 0 to ∞ and c = nα, we obtain 

∞ 0

t

k−1 −nαt

e

k−1 dt = nα



∞ 0

t k−2 e−nαt dt,

150

4 Epidemic Models

which leads by induction to the formula 

∞

(k − 1)! 1 · . (nα)k−1 nα

t k−1 e−nαt dt =

0

(4.32)

Combination of (4.30), (4.32) gives 

∞ 0

P (τ )dτ =

n 1 1 = . nα α k=1

The mean infective period is independent of n but a larger value of n gives a smaller variance. The limiting case as n → ∞ is an infective period of fixed length 1/α. In practice, infective periods often are bunched closer together than would be predicted by an exponential distribution and it is common to choose a gamma distribution with a value of n corresponding to the observed  ∞ distribution. We will need to use the value of 0 e−λτ P (τ )dτ. To evaluate this integral, we have 

∞

e

−λτ

0

(nα)k−1 uk (τ )dτ = (k − 1)!



∞

0

τ k−1 e−(λ+nα)τ dτ =

(nα)k−1 , (λ + nα)k

using (4.32). From this we obtain 

0

∞

e

−λτ

n−1 n−1 nα 1 (nα)k−1 . = P (τ )dτ = (λ + nα)k (λ + nα) λ + nα

(4.33)

k=0

k=0

But this is a geometric series, whose sum is 1 λ + nα



n

nα λ+nα nα 1 − λ+nα

1−

=

1−



nα λ+nα

λ

n

.

We now have the formula 

∞

0

e−λτ P (τ )dτ =

1−



nα λ+nα

λ

n

(4.34)

.

The limiting case as n → ∞ of a gamma distribution is a distribution P (t) = 1

(0 ≤ t ≤

1 ), α

P (t) = 0

(t >

1 ), α

4.7 Interpretation of Data and Parametrization

151

for which 

∞

0

e−λt P (t)dt =

1 − e−λ/α . γ

Then 1 α+

γ n




1 ), α

4.7 Interpretation of Data and Parametrization

153

Then 1 α+


er /κ. α(1 − e−r/α )

It is possible to show that the basic reproduction number corresponding to fixed exposed and infective periods is always larger than the basic reproduction number corresponding to exponentially distributed exposed and infectious periods, and the basic reproduction number corresponding to exposed and infectious periods both having gamma distributions is between these two values.

4.7.3 Mean Generation Time In a simple epidemic model, if we know the contact rate and the infective period, then we may be able to determine the basic reproduction number. As we have seen in the previous section, knowledge of the initial exponential growth rate, which may sometimes be observed experimentally leads to information about the contact rate. Another quantity which may sometimes be observed experimentally, is the mean generation time, meaning roughly the mean time between a primary case and a

4.7 Interpretation of Data and Parametrization

157

secondary case caused by this primary case. This quantity has been given various definitions, not all equivalent. Another term often used to describe this idea is the serial interval. The mean generation time is defined [18, 40] as the mean time from infection to the onset of infectiousness. It may be estimated from data early in a disease outbreak [40, 44, 45]. If we assume that the rate of secondary infections caused by an individual is proportional to the infectiousness of the individual [37], then the rate of secondary infections caused at infection age τ is proportional to A(τ ). This assumption would be valid only early in an epidemic when the depletion of the susceptible population would be negligible and the number of infectives is a sufficiently small fraction of the total population size that a susceptible is at risk of infection only from a single infective. Thus the distribution of new infections caused at infection age τ is A(s) ∞ , 0 A(s)ds

whose mean is

∞

sA(s)ds TG = 0 ∞ , 0 A(s)ds

(4.46)

and this is the mean generation time. We should note that this describes the time interval between successive infections, but what may be observed is the time interval between development of clinical symptoms, which is not the same thing, but we may hope that it is a reasonable approximation. In order to calculate the mean generation time for the general SEIR model (4.45), we use (4.46) with A(s) = AI (s) = −



s 0

PE′ (s − u)PI (u)du.

As we have seen in the calculations for the model (4.45), ∞ 0 PI (s)ds. We have also, 

∞

0

sA(s)ds = − =−





0

0

∞ s 0

sPE′ (s − u)PI (u)dsdu

∞  ∞ u

∞ 0

A(s)ds

(4.47)

 sPE′ (s − u)ds PI (u)du.

But −



∞ u

′ sEQ (s − u)ds = −



∞ 0

(u + v)PE′ (v)dv = u +



0

=

∞

PE (v)dv,

158

4 Epidemic Models

using integration by parts. Thus (4.47) becomes 

∞

0

sA(s)ds =



∞ 0

uPI (u)du +



∞

PI (u)du

0



∞

PE (v)dv.

0

This shows that the mean generation time for the SEIR model (4.45) is ∞

uPI (u)du + TG = 0 ∞ 0 PI (u)du



∞

PE (u)du.

0

Note that this is the sum of the mean period of stay in the exposed stage and a quantity depending on the infective period distribution. It is easy to calculate that in the case of an exponential distribution with mean 1/α this quantity is 1/α, the mean infective period, while in the case of a constant infectious period of length 1/α this quantity is 1/2α, half the mean infective period. In general, for an SEIR model the mean generation time is equal to the sum of the mean exposed period and a quantity depending on the distribution of the infective period. It is convenient to define the mean exposed time TE and the mean infective time TI . For the general SEIR model (4.45) that we have been studying, we have TE =

1 , κ

TI =

1 . α

For exponentially distributed exposed and infective periods, TG = TE + TI , and we may write R0 = 1 + r

1 1 1 + r + r2 = 1 + rTE + rTI + r 2 TE (TG − TE ). κ α κα

For exposed and infective periods of fixed length, 1 TG = TE + T I , 2 and we have R0 =

r r/κ αe 1 − e−r/α

=

r r/κ+r/2α αe er/2α − e−r/2α

=

r rTG αe

2 sinh(r/2α)

.

Since sinh x > x for x > 0, we obtain the bound R0 < erTG .

(4.48)

4.8 *Effect of Timing of Control Programs on Epidemic Final Size

159

For a model with an exposed period having a gamma distribution with parameter m and an infective period having a gamma distribution with parameter n it is possible to show that the corresponding reproduction number, which we denote by (m,n) R0 , is m

r +1 r κm (m,n) . (4.49) R0 =  r + 1)−n α 1 − ( αn

It is possible to show that R0(m,n) is an increasing function of both m and n, and that as m, n → ∞, R0(m,n) approaches the reproduction number corresponding to exposed and infective periods of fixed length.

4.8 *Effect of Timing of Control Programs on Epidemic Final Size To explore the influence of seasonally forced transmission in disease control and the epidemic final size of influenza, we consider an extension of the standard SIR epidemic model by incorporating a periodic function c(t) for the transmission rate as well as drug treatment and/or vaccination. When model parameters vary with time, analytic results on final size will be very difficult to obtain. Most results are based on numerical simulations. The example presented here with periodic transmission rate examines the potential negative effect of vaccination and/or treatment uses depending on the timing of implementation. These scenarios are based on the consideration that vaccines and antiviral drugs may not be available at the beginning of an epidemic. We focus on three important measures when assessing the effect of a control program: (a) peak size of the epidemic curve (the maximum number of infections during the course of a pandemic); (b) peak time (the time at which the peak occurs); and (c) final size (the total number of infections at the end of a pandemic). Main objectives of effective control strategies should include lowering the peak size to keep demand for facilities below available supply, lowering the final size to reduce morbidity, and delaying the peak to provide more time for response. Unlike in the case of constant parameters, where vaccination and treatment will always help reduce morbidity, the case of a periodic transmission rate c(t) may generate non-intuitive results. That is, the model can exhibit outcomes in which an increased use of vaccination or antiviral drugs will lead to a higher morbidity. To demonstrate this, we first consider the following epidemic model with timedependent infection rate SI dS = −c(t) dt N SI dI = c(t) − αI, dt N

(4.50)

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4 Epidemic Models

where

with the initial conditions

  c(t) = c0 1 + ǫ cos(2π t/365)

S(t0 ) = (1 − ϕ0 )(N − I0 ), I (t0 ) = I0 , R(t0 ) = 0.

(4.51)

The parameter ǫ denotes the magnitude of the seasonal forcing and the period is assumed to be 1 year. We do not include vaccinated individuals in the R class because the value of R(t) at the end of the disease outbreak will be used to measure the final epidemic size. Notice that the initial time t0 represents the time at which the vaccination program starts. Some simulation results are presented in Fig. 4.6.

Fig. 4.6 Plots of epidemic curves and cumulative infections for different t0 values. These plots show the dramatic effect that the start time of an epidemic (t0 ) can have on its progression in a periodic environment given by the SIR model (4.50) with initial conditions (4.51). From the plots in (a) and (c) we observe that when t0 = 30, vaccination reduced both the peak size and the peak time although there is not much delay in the peak time. However, plots in (b) and (d) show that, although the size of the first epidemic peak is reduced, a second (and higher) peak is also generated. More significantly, the final epidemic size is increased, showing the sensitivity of the behavior of the epidemic to t0

4.10 Some Warnings

161

In Fig. 4.6, both the epidemic curve and curve representing the cumulative infections are plotted. For the transmission function c(t), the parameter values used are those of an H1N1-like disease with c0 = 0.5 and ǫ = 0.35. We assume 1/α = 3 days. These values are also used in other figures except when specified differently. The figure shows two sets of simulations corresponding to two different times t0 of initial introduction of pathogens. One case is for t0 = 30 and the other is for t0 = 40, and for demonstration purposes the vaccination level is chosen to be ϕ0 = 0.1. Some interesting observations can be made from Fig. 4.6. We can first compare the outcomes between the model without vaccination (the solid curves) and the model with vaccination (the dashed curves). It demonstrates the dependence of the model outcomes on the time of introduction (t0 ) of initial infections. Particularly, we observe that although the peak and final sizes are both decreased for the case of t0 = 30, the case of t0 = 40 differs significantly. Although the first wave is reduced, the second wave is also generated; and moreover, the final epidemic size is even higher than that without vaccination, demonstrating a scenario in which the use of vaccine may have an adverse effect.

4.9 Directions for Generalization A fundamental assumption in the model (4.1) is homogeneous mixing, that all individuals are equivalent in contacts. A more realistic approach would include separation of the population into subgroups with differences in behavior. For example, in many childhood diseases the contacts that transmit infection depend on the ages of the individuals, and a model should include a description of the rate of contact between individuals of different ages. Other heterogeneities that may be important include activity levels of different groups and spatial distribution of populations. Network models may be formulated to include heterogeneity of mixing, or more complicated compartmental models can be developed. An important question which should be kept in mind in the formulation of epidemic models is the extent to which the fundamental properties of the simple model (4.1) carry over to more elaborate models.

4.10 Some Warnings An actual epidemic differs considerably from the idealized model (4.1) or (4.17). Some notable differences are: 1. When it is realized that an epidemic has begun, individuals are likely to modify their behavior by avoiding crowds to reduce their contacts and by being more careful about hygiene to reduce the risk that a contact will produce infection. 2. If a vaccine is available for the disease which has broken out, public health measures will include vaccination of part of the population. Various vaccination

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4 Epidemic Models

strategies are possible, including vaccination of health care workers and other first line responders to the epidemic, vaccination of members of the population who have been in contact with diagnosed infectives, or vaccination of members of the population who live in close proximity to diagnosed infectives. 3. Isolation may be imperfect; in-hospital transmission of infection was a major problem in the SARS epidemic. 4. In the SARS epidemic of 2002–2003, in-hospital transmission of disease from patients to health care workers because of imperfect isolation accounted for many of the cases. This points to an essential heterogeneity in disease transmission which must be included whenever there is any risk of such transmission.

4.11 *Project: A Discrete Model with Quarantine and Isolation Project 1 This project concerns a general single-outbreak discrete epidemic model involving arbitrarily distributed stage-duration distributions for the latent and the infectious stages. Clarity is added to model derivation by making use of a probabilistic perspective. Hence, we let X and Y represent the time an individual spends in latent (E, Q) and infectious (I, H ) classes, respectively. Similarly, we denote by Z the time at which an exposed individual is quarantined (from E to Q), and W the time at which an infected individual is isolated (from I to H ). X, Y , Z, and W must take values on {1, 2, 3, . . . } and their dynamics are assumed to be governed by underlying probabilistic processes. It is understood that the distribution of waiting time in each of the above classes can be described by probability distributions associated with the random variables X, Y , Z, and W : pi = P(X > i) probability of remaining latent i steps after entering E, qi = P(Y > i) probability of remaining infectious i steps after entering I , ki = P(Z > i) probability of not quarantined i steps after entering E, li = P(W > i) probability of not isolated i steps after entering I . (4.52) Assume that p0 = q0 = k0 = l0 = 1, meaning that the latency, infectious, quarantine , and isolation periods last at least one time step. For ease of description, we introduce the following notation: An Bn Cn Dn Fn

the input to E at time n (new infections), the input to I at time n, the input to Q at time n, the input to H from Q at time n, (= Bn + Dn ) the total input to infectious class at time n.

A transition diagram using the above notation is shown in Fig. 4.7.

4.11 *Project: A Discrete Model with Quarantine and Isolation Fig. 4.7 Transition diagram for the model with arbitrarily distributed stage durations

S

E

163

I R

Q

H

Using the above notation the model with arbitrarily distributed stage durations can be written as: Sn+1 = Sn Gn ,

β

Gn = e− N [In +(1−ρ)Hn ] ,

En+1 = An+1 + An p1 k1 + · · · + A1 pn kn + A0 pn+1 kn+1 , Qn+1 = An p1 (1 − k1 ) + · · · + A1 pn (1 − kn ) + A0 pn+1 (1 − kn+1 ),

(4.53)

In+1 = Bn+1 + Bn q1 l1 + · · · + B1 qn ln , Hn+1 = Fn+1 + Fn q1 + · · · + F1 qn − In+1 ,

n = 0, 1, 2, · · · ,

where An , Bn , Cn , and Dn are given by An+1 = Sn (1 − Gn ) = Sn − Sn+1 , n ≥ 0, with A0 = E0 , Bn+1 = An (1 − p1 ) + An−1 (p1 − p2 )k1 + · · · + A1 (pn−1 − pn )kn−1 +A0 (pn − pn−1 )kn , with B0 = 0, k0 = 1,

(4.54)

Cn+1 = En − En+1 + An+1 − Bn+1 , with C0 = 0, Dn+1 = Qn − (Qn+1 − Cn+1 ). The initial conditions in model (4.53) are S0 , E0 > 0 and I0 = Q0 = H0 = R0 = 0. Question 1 Show that the control reproduction number Rc is given by Rc = RI + RI H + RQH ,

(4.55)

where RI = βTEk DIl , RI H = β(1 − ρ)TEk (DI − DIl ),

(4.56)

RQH = β(1 − ρ)(1 − TEk )DI . Here, DE = E(X) and DI = E(Y ) are the mean sojourn times in the exposed and infectious stages, respectively. DEk and DIl denote the “quarantine-adjusted”

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4 Epidemic Models

and “isolation-adjusted” mean sojourn times in the exposed and infective stages, respectively, given by DEk = E(X ∧ Z) = DIl = E(Y ∧ W ) =

∞

j =0 ∞

j =0

P(X ∧ Z > j ) = 1 +

P(Y ∧ W > j ) = 1 +

∞

pj k j ,

j =1 ∞

qj lj .

j =1

TEk denotes the proportion of individuals in the E class who become infective before going to the Q class, i.e., the event E → I occurs before E → Q. This quantity is given by TEk = P(X ≤ Z) =

∞ (pj −1 − pj )kj −1 . j =1

The three quantities in (4.56) represent the stage-specific reproduction numbers representing, respectively, the average number of secondary infections produced by individuals in the I stage, in the H stage with quarantine (QH ) and without (I H ). Question 2 Let Rc be as given in (4.55). Show that the epidemic final size generated by the dynamics of model (4.53) satisfies the following final size relationship: log

  S∞ S0 Rc . = 1− S∞ N

(4.57)

Question 3 Consider the case where the distributions in (4.52) are geometric. Assume that X ∼ Geom(α), Y ∼ Geom(δ), Z ∼ Geom(γ ), and W ∼ Geom(σ ). That is, pi = α i ,

qi = δ i ,

ki = γ i ,

li = σ i ,

i = 0, 1, 2, . . . .

In this case, the transition diagram 4.7 can be replaced by that shown in Fig. 4.8

Fig. 4.8 Transition diagram for the model with geometric stage distributions

S

E

I R

Q

H

4.11 *Project: A Discrete Model with Quarantine and Isolation

165

Show that under this assumption, the general model (4.53) simplifies to Sn+1 = Sn Gn En+1 = (1 − Gn )Sn + αγ En Qn+1 = α(1 − γ )En + αQn

(4.58)

In+1 = (1 − α)En + δσ In Hn+1 = (1 − α)Qn + δ(1 − σ )In + δHn ,

n = 0, 1, 2, 3, · · · .

Note that the system (4.58) has the same form as the usually considered model with constant probabilities for transitions between stages. We refer to this model as the geometric distribution model (GDM). The questions below concern the GDM (4.58). (a) Determine the quantities DE , DI , DEk , DIl , and TEk . (b) Compute RI , RI H , RQH , and the control reproduction number Rc . Question 4 We can examine how different distribution assumptions may influence the values of the reproduction number Rc given in (4.55) and (4.56). Particularly, the reduction in Rc by quarantine and isolation can be different under different distribution assumptions. To examine this, we can consider two specific distributions, geometric distribution assumption (GDA) and Poisson distribution assumption (PDA). For the purpose of comparison, we take Xg and Xp with mean μ1 and Yg and Yp with mean μ2 (the subscripts g and p represent geometric and Poisson, respectively). The quarantine (Z) and isolation (W ) are assumed to have geometric distributions with parameters γ or σ , respectively. Consider the parameter values β = 0.75, ρ = 0.95, μ1 = 5, μ2 = 10. Compare two strategies: Strategy I corresponds to γ = 0.5 and σ = 0.8 and Strategy II corresponds to γ = 0.8 and σ = 0.5. Figure 4.9 plots the control reproduction number Rc vs γ or σ . The left figure is for Rc,g (i.e., when X and Y are geometric distributions) and the right figure is for Rc,p (i.e., when X and Y are Poisson distributions). Figure 4.9 shows reduction in Rc under Strategy I (represented by a dot on the dashed curve) and Strategy II (represented by a solid square on the solid curve). (a) Reproduce Fig. 4.9. (b) Under the GDA, which of the strategies I and II is more effective in reducing the control reproduction number? (c) Under the PDA, which of the strategies I and II is more effective in reducing the control reproduction number? (d) Do the two distribution assumptions generate the same assessment? Why or why not? (e) The formula (4.57) holds for arbitrarily distributed latent and infectious periods. Because reductions of Rc by quarantine and isolation depend on distribution assumptions, compare the effects of these control measures on the epidemic final size under the GDA and PDA.

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4 Epidemic Models

Fig. 4.9 Comparison of the GDM and PDM in terms of their evaluations on various control measures when the corresponding control reproduction numbers, Rc,g and Rc,p are used. The two control strategies considered are represented by γ = 0.5 and σ = 0.8 (labeled with a dot on the dashed curve) and by γ = 0.8 and σ = 0.5 (labeled with a solid square on the solid curve)

Project 2 Consider the epidemic model (4.26), which includes general distributions for the latent and infectious stages with survival function Pi (s) (i = E, I ), as well as a control measures quarantine and isolation represented by parameters ψ and ϕ. The model also assumes that the isolation is 100% effective, i.e., the isolated individuals do not contribute to the disease transmission. Let qi (s) = −Pi′ (s) (i = E, I ). The corresponding reproduction number is Rc (ψ, ϕ) = βN



∞ 0

e−ψy qE (y)dy



∞

e−ϕu PI (u)du,

(4.59)

0

or in terms of the Laplace transforms of qE and PI Rc (ψ, ϕ) = βN LqE (ψ)LPI (ϕ).

(4.60)

In the more general case of imperfect isolation, the isolation may not be 100% effective. Let σ denote the isolation efficiency for isolated individuals with 0 ≤ σ ≤ 1. Then the effective transmission rate for the isolated (or hospitalized) individuals (H ) is reduced to (1 − σ )β. In this case, the S equation in (4.26) becomes S ′ = −βS[I + (1 − σ )H ].

(4.61)

∞ Question 1 Let μI denote the mean infective period, i.e., μI = 0 PI (s)ds. Show that the formula (4.60) can be extended to be a function of ψ, ϕ, and σ as below: Rc (ψ, ϕ, σ ) = (1 − σ )βNLqE (ψ)μI + σβNLqE (ψ)LPI (ϕ).

(4.62)

4.12 Project: Epidemic Models with Direct and Indirect Transmission

167

Question 2 Ifthe stage durations are both gamma distributed with shape parameters kE ≥ 1 and kI ≥ 1, i.e., qi (s, μi , ki ) =

1 ki  ki ki −1 − μki s s e i , s > 0, Γ (ki ) μi μi

i = E, I.

(a) Show that in the absence of quarantine (i.e., φ = 0) the control reproduction number is   −kI  μI σβN Rc (0, ϕ, σ ) = (1 − σ )βNμI + 1− 1+ . ϕ ϕ kI

(4.63)

(b) Let βN = 0.5. Plot Rc (0, ϕ, σ ) given in (4.63) for various sets of values of kE = kI = k, μI , and σI . For example, k = 1, 2, 4, μI = 3, 5, 8, and σ = 0.1, 0.5, 0.8, 1. Summarize the observations from these plots in terms of the dependence of Rc on the shape parameter, mean period of infection, and isolation efficiency. (c) Let kE = kI = k. Note that k = 1 corresponds to an exponentially distributed stage, which is the most commonly assumed distribution. For the two values k = 1 and 4, draw the contour plots of Rc (0, ϕ, σ ) in the (ϕ, σ ) plane showing a few contour curves including the one corresponding to Rc (0, ϕ, σ ) = 1. Other parameter values are βN = 0.5, μI = 3, σ = 1. Discuss the observed difference from the contour plots between the cases of k = 1 and k = 4 in terms of the threshold values of ϕ for which Rc (0, ϕ, σ ) = 1.

4.12 Project: Epidemic Models with Direct and Indirect Transmission Some diseases may be spread in more than one manner. For example, cholera may be spread by person to person transmission but may also be transmitted indirectly through a pathogen released by infectives through a medium such as contaminated water [38, 41]. Consider an epidemic model with direct (person to person) and indirect (through a medium such as contaminated water) transmission. To a simple SI R model we add a pathogen B shed by infectives. We assume that the infectivity of the pathogen is proportional to its concentration, suggesting mass action transmission. The resulting model is S ′ = −β1 SI − β2 SB

I ′ = β1 SI + β2 SB − γ I ′

R = γI

B ′ = rI − δB,

(4.64)

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4 Epidemic Models

with initial conditions S(0) = S0 ,

I (0) = I0 ,

B(0) = B0 ,

in a population of constant total size N = S0 + I0 , with R(0) = 0. In general, N = S + I + R. In this model r represents the rate at which an infectious individual sheds pathogen and δ represents the rate at which the pathogen loses infectivity. Question 1 Show that the basic reproduction number R0 =

rβ2 N β1 N + . γ γδ

In this expression, the first term represents secondary infections caused directly by a single infective introduced into a wholly susceptible population, infecting βN susceptibles in unit time for a time period 1/γ . The second term represents secondary infections caused indirectly through the pathogen since a single infective sheds a quantity r of pathogen in unit time for a time period 1/γ and this pathogen infects βN susceptibles in unit time for a time period 1/δ. Question 2 Derive the final size relation log

   S0 B0 S∞ β1 Nr 1− + β1 = β1 + S∞ γ δ N δ   S∞ B0 + β1 . = R0 1 − N δ

(4.65)

This implies S∞ > 0. In order to cover such generalizations of the model (4.1) as multiple infective stages and arbitrary distributions of stay in a stage, we give an age of infection model S ′ (t) = −S(t)[β1 ϕ(t) + β2 B(t)]  t ϕ(t) = ϕ0 (t) + [−S ′ (t − τ )]P (τ )dτ

(4.66)

0

B(t) = B0 (t) +



t

0

rϕ(t − τ )Q(τ )dτ.

In this model, ϕ(t) represents the total infectivity of individuals with age of infection t, ϕ0 (t) represents the total infectivity at time t of individuals who were already infected at time t = 0, B0 (t) represents the pathogen concentration at time t remaining from pathogen concentration that was already present at time t = 0, P (τ ) represents the mean infectivity of individuals at age of infection τ , normally the product of the fraction of infectives still infective at age of infection τ and the

4.12 Project: Epidemic Models with Direct and Indirect Transmission

169

relative infectivity at that infection age, and Q(τ ) represents the fraction of pathogen remaining τ time units after having been shed  ∞ by an infective. The function Q is monotone non-increasing with Q(0) = 1, 0 Q(τ )dτ < ∞. Since infectivity of an individual may depend on the age of infectionof the individual, the function P ∞ is not necessarily non-increasing, but we assume 0 P (τ )dτ < ∞. Question 3 Show that for the model (4.66), the basic reproduction number is R0 = β1 N



∞ 0

P (τ )dτ + rβ2 N



∞

P (τ )dτ

0



∞

Q(τ )dτ.

0

In this expression, the first term represents new infection transmitted directly by a single infectious individual inserted into a totally susceptible population, while the second term represents secondary infections caused by this individual indirectly through shedding of pathogen. Question 4 Obtain the final size relation    ∞ S0 − S∞ S0 + β1 ϕ0 (t)dt = R0 log S∞ N  ∞0  ∞  Q(t)dt ϕ0 (t)dt + β2 +rβ2 0

0

∞

(4.67) B0 (t)dt.

0

If all infections at time zero have infection age zero, then ϕ0 (t) = [N − S0 ]P (t),



∞ 0

ϕ0 (t)dt = [N − S0 ]



∞

P (t)dt,

0

and if the entire pathogen concentration at time zero has infection age zero, then B0 (t) = B0 Q(t),



0

∞

B0 (t)dt = B0



∞

Q(t)dt

0

with some constant B0 . In this case, the final size relation (9.10) takes the form log

   ∞ S0 S∞ + β2 B0 Q(t)dt. = R0 1 − S∞ N 0

(4.68)

The final size relation has a term arising from an initial pathogen concentration that tends to decrease S∞ . In general, because Q is monotone non-increasing, 

∞ 0

B0 (t)dt ≤ B0



∞ 0

Q(t)dt.

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4 Epidemic Models

If P is monotone non-increasing, 

∞ 0

ϕ0 (t)dt ≤ [N − S0 ]



∞

P (t)dt.

0

If P is not monotone, which may occur, for example, if there is an exposed stage followed by an infective stage with higher infectivity, this is not necessarily true. However, if there are no infectives initially, so that the epidemic is started by the pathogen, then ϕ0 (t) = 0 and S0 = N . Then (4.67) remains valid without the need to assume that P is monotone. These results have been established only for a constant rate of pathogen shedding. If the rate of pathogen depends on the age of infection, the equation for B in the model (4.66) should be replaced by an equation  t B(t) = B0 (t) + r(t − τ )ϕ(t − τ )Q(τ )dτ. 0

It is not possible to treat the corresponding model as an age of infection model, but we can view it as a staged progression model. We consider an epidemic with progression from S through k infected stages I1 , I2 , · · · , Ik , as analyzed in [7], but with the addition of a pathogen. We assume that in stage j the relative infectivity is εj , the distribution of stay in the stage is ∞ given by Pj with Pj (0) = 1, 0 Pj (t)dt < ∞, and Pj monotone non-increasing, so that the infectivity of an individual in stage j is Aj (τ ) = εj Pj (τ ). There are no disease deaths and the total population size N is constant. We assume initial conditions S(0) = 0,

I1 (0) = I0 ,

I2 (0) = I3 (0) = · · · = Ik (0) = 0,

R(0) = 0.

The total infectivity is given by ϕ(t) =

k

εj Ij (t).

j =1

We let Bj (t) be the quantity of pathogen shed by infectives in the stage Ij and let Qj denote the  ∞distribution of stay of pathogen shed by infectives in this stage, with Qj (0) = 1, 0 Qj (t)dt < ∞, and Qj monotone non-increasing. We let rj be the shedding rate in this stage. We also define the total quantity of pathogen, B(t) =

k

Bj (t).

j =1

Then Bj (t) = Bj0 (t) +



t 0

rj Ij (t − τ )Qj (τ )dτ.

(4.69)

4.13 Exercises

171

Question 5 Show that the basic reproduction number is R0 = β1 N

k j =1

εj



∞

Pj (t)dt + β2 N

0

k

rj

j =1



∞

Pj (t)dt

0



∞

Qj (t)dt.

0

Question 6 For simplicity, we assume that all individuals infected at time zero have infection age zero for t = 0, and also that there is a new quantity of pathogen B0 introduced at time zero, so that B0 (t) = B0 Q(t). Obtain the final size relation 

k

log

S0 = β1 εj S∞ j =1

+β2

k

rj

0



Pj (t)dt[N − S∞ ]

∞

Pj (t)dt

0

j =1

+β2 B0

∞



∞



0

∞

Qj (t)dt[N − S∞ ]

Q(t)dt

(4.70)

0

   ∞ S∞ + β2 B0 = R0 1 − Q(t)dt. N 0 References [8]. Other sources of information about cholera include [1, 2, 11, 21, 25, 38, 41].

4.13 Exercises 1. Show that it is not possible for a major epidemic to develop unless at least one member of the contact network has degree at least 3. 2. What is the probability of a major epidemic if every member of the contact network has degree 3? 3. Show that if G1 (0) = 0, then (a) (b) (c) (d)

G′1 (0) ≤ 0 G1 (z) < z for 0 ≤ z ≤ 1 z∞ = 0 R0 > 1

4. Consider a truncated Poisson distribution, with  −c k e c k! , k ≤ 10, pk = 0, k > 10. Estimate (numerically) the probability of a major epidemic if c = 1.5.

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4 Epidemic Models

5. Show that the probability generating function for an exponential distribution, given by pk = (1 − e−1/r )e−k/r , is G0 (z) =

1 − e−1/r . 1 − ze−1/r

6. A power law distribution is given by pk = Ck −α . For

what values of α is it possible to normalize this (i.e., choose C to make pk = 1)? 7. Consider a network in which the contacts between members of the population are random. Assume that the fraction of vertices having degree k, or the probabilities pk are given by the Poisson distribution pk = ec ck /k!, where c = R0 = 3. Let P denote the probability of a minor outbreak. (a) Determine the equation that can be used to solve for P. (b) Find P. (b) Find the probability of a major epidemic. 8. Compare the qualitative behaviors of the models S ′ = −βSI,

I ′ = βSI − αI,

and S ′ = −βSI,

E ′ = βSI − κE,

I ′ = κE − αI,

with βN = 1/3,

α = 1/6,

κ = 1/2,

S(0) = 999,

I (0) = 1.

These models represent an SIR epidemic model and an SEIR epidemic model respectively with a mean infective period of 6 days and a mean exposed period of 2 days. Do numerical simulations to decide whether the exposed period affects the behavior of the model noticeably. 9. Consider three basic epidemic models—the simple SIR model, S ′ = −βSI,

I ′ = βSI − αI,

4.13 Exercises

173

the SEIR model with some infectivity in the exposed period, S ′ = −β[S(I + εE],

E ′ = β[S(I + εE] − κE, I ′ = κE − αI,

and the SIR model with treatment, S ′ = −βS(I + δT ),

I ′ = βS(I + δT ) − (α + ϕ)I,

T ′ = ϕI − ηT . Use the parameter values βN =

1 , 3

α=

1 , 4

ε=

1 , 2

κ=

1 , 2

δ=

1 , 2

η=

1 , 4

ϕ = 1,

and the initial values S(0) = 995,

E(0) = 0,

I (0) = 5,

T (0) = 0.

For each model, (a) Calculate the reproduction number and the epidemic final size. (b) Do some numerical simulations to obtain the epidemic size by determining the change in S, the maximum number of infectives by measuring I , and the duration of the epidemic. 10. Consider the treatment model (4.12). Assume that βN = 860, δ = 0.5, η = 1/10, and S0 = N − 1. (a) Explore numerically how the final epidemic size can be affected by the treatment effort (measured by the parameter γ ). Plot the relationship. (b) Determine the final epidemic size without treatment. (c) Is it possible to reduce the final size by 50% by increasing γ ? If yes, find that γ value. 11. For the model (4.15), what should be the constraints on parameter values so that passage from E to Q to J to R is at least as rapid as the passage from E to I to R, and the passage from I to J to R is at least as rapid as the passage from I to R? 12. Consider the model (4.15) with quarantine and isolation. Assume that βN = 860, κE = 1/4, αI = 1/10, εE = 0.4, εQ = 0.1, εJ = 0.6.

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(a) In the absence of control (i.e., γQ = γJ = κQ = αJ = 0), what is the final epidemic size? (b) Plot the final size as a function of γQ and γJ . (c) Is it possible to reduce the final size by 50% by increasing γQ and/or γJ ? If yes, identify one pair of values of γQ and γJ to achieve the goal. 13. Formulate an SEI RT model, that is, a model with both an exposed class and a treatment class (a) Draw a flow chart and calculate the basic reproduction number (b) Determine the final size relation 14. Consider an SIR model in which a fraction θ of infectives is isolated in a perfectly quarantined class Q with standard incidence (meaning that individuals make a contacts in unit time of which a fraction I /(N − Q) is infective), given by the system S ′ = −βN S I ′ = βN S

I , N −Q

I − (θ + α)I, N −Q

Q′ = θ I − γ Q,

R ′ = αI + γ Q.

(a) Find the equilibria. (b) Find the basic reproduction number R0 . (c) For parameters, take α = 0.1, θ = 1, 2, 4, γ = 0.2. Sketch the phase plane of the system and observe what is happening. 15. Isolation/quarantine is a complicated process because we don’t live in a perfect world. In hospitals, patients may inadvertently or deliberately break from isolation and in the process have casual contacts with others including medical personnel and visitors. Taking this into account, we are led to the model S ′ = −βN S I ′ = βN S

[I + ρτ Q] , N − σQ

[I + ρτ Q] − (θ + α)I, N − σQ

Q′ = θ I − γ Q,

R ′ = αI + γ Q.

(a) Determine all the parameters in the system and define each parameter. (b) Show that the population is constant.

4.13 Exercises

175

(c) Find all equilibria. (d) Find the reproduction number R0 . (e) Describe the asymptotic behavior of the model, including its dependence on the basic reproduction number. 16. Formulate a model analogous to (4.12) for which treatment is not started immediately, but begins at time τ > 0. Can you say anything about the dependence of the reproduction number on τ ? 17. In the simple SI R model (4.1) it is assumed that all effective contacts between susceptibles and infectives lead to new infections. Suppose, however, that only a fraction δ of contacts by infectives actually transmit disease and only a fraction σ of contacts of susceptibles actually lead to new infections. This would lead to a model S ′ = −δσβSI

I ′ = δσβSI − αI.

Find the basic reproduction number and the final size relation for this model. 18. Consider the model in Sect. 4.4 and assume that P (τ ) has the value 1 if 0 ≤ τ ≤ T and 0 otherwise. (a) Calculate the basic reproduction number (b) Plot the final epidemic size for several different values of T . 19. Consider the model in Sect. 4.4 and assume that P (τ ) follows a gamma distribution with mean equal to 10 and shape parameter equal to n. (a) Plot final epidemic size for n = 1, 3, 5, 7, 9. (b) Does the final size change with n? If yes, what is the observed pattern of the change? 20. The quarantine–isolation model (4.26) is not quite in the age of infection form that we have been studying in this section because of the initial terms in the equations. Transform it to age of infection form by incorporating the initial terms into infinite integrals and calculate the control reproduction number from this form. 21. Determine the initial exponential growth rate for the model (4.11) by interpreting it as an age of infection model. 22. Determine the initial exponential growth rate for the model (4.12) by interpreting it as an age of infection model. 23. Determine the initial exponential growth rate for the general treatment model (4.24) of Sect. 4.5. 24. Consider the model (4.23) with A(τ ) = e−0.1τ . (a) Determine the critical value of ac such that there is an epidemic if a > ac and no epidemic if a < ac . (b) Is there an epidemic for a = 1.5? If yes, find the exponential grow rate of the epidemic.

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25. 26. 27. 28.

Find the solution of the system (4.29). Verify the relation (4.30). Verify the Eq. (4.32). Determine the mean generation time for an SEIR model with an infectious period having a gamma distribution with parameter n. 29. Establish the relation (4.49). 30. Consider the model (4.50) with initial conditions given in (4.51) with t0 = 30. The parameter values c0 , ε, and α are the same as those used in Fig. 4.6. Make plots similar to those in Fig. 4.6 under the following scenarios. (a) Plot the epidemic curves for three values of vaccination fraction p: 0.05, 0.1, and 0.15. Compare the shape of each epidemic curve with that for the case of no vaccination (i.e., p = 0). (b) Plot the cumulative cases for p = 0.05, 0.1, and 0.15. Compare each case with the case of no vaccination. (c) How many epidemic waves do you observe in each of the three cases? (d) Is it true that the higher the fraction of people vaccinated the lower the final size will be? 31. Repeat Exercise 30 but for t0 = 40. Describe the qualitative differences in epidemic curves and cumulative cases between the case t0 = 30 and the case t0 = 40? What are the implications of these differences for public health policy decisions?

References 1. Alexanderian, A., M.K. Gobbert, K.R. Fister, H. Gaff, S. Lenhart, and E. Schaefer (2011) An age-structured model for the spread of epidemic cholera: Analysis and simulation, Nonlin. Anal. Real World Appl. 12: 3483–3498. 2. Andrews, J.R. & and S. Basu (2011) Transmission dynamics and control of cholera in Haiti: an epidemic model, Lancet 377: 1248–1255. 3. Arino, J., F. Brauer, P. van den Driessche, J. Watmough & J. Wu (2006) Simple models for containment of a pandemic, J. Roy. Soc. Interface, 3: 453–457. 4. Arino, J. F. Brauer, P. van den Driessche, J. Watmough& J. Wu (2007) A final size relation for epidemic models, Math. Biosc. & Eng. 4: 159–176. 5. Arino, J., F. Brauer, P. van den Driessche, J. Watmough & J. Wu (2008) A model for influenza with vaccination and antiviral treatment, Theor. Pop. Biol. 253: 118–130. 6. Bansal, S., J. Read, B. Pourbohloul, and L.A. Meyers (2010) The dynamic nature of contact networks in infectious disease epidemiology, J. Biol. Dyn., 4: 478–489. 7. Brauer, F., C. Castillo-Chavez, and Z. Feng (2010) Discrete epidemic models, Math. Biosc. & Eng. 7: 1–15. 8. Brauer, F., Z. Shuai, & P. van den Driessche (2013) Dynamics of an age-of-infection cholera model, Math. Biosc. & Eng. 10:1335–1349. 9. Brauer, F., P. van den Driessche and J. Wu, eds. (2008) Mathematical Epidemiology, Lecture Notes in Mathematics, Mathematical Biosciences subseries 1945, Springer, Berlin - Heidelberg - New York.

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35. Riley, S., C. Fraser, C.A. Donnelly, A.C. Ghani, L.J. Abu-Raddad, A.J. Hedley, G.M. Leung, L-M Ho, T-H Lam, T.Q. Thach, P. Chau, K-P Chan, S-V Lo, P-Y Leung, T. Tsang, W. Ho, K-H Lee, E.M.C. Lau, N.M. Ferguson, & R.M. Anderson (2003) Transmission dynamics of the etiological agent of SARS in Hong Kong: Impact of public health interventions, Science 300: 1961–1966. 36. Roberts, M.G. and J.A.P. Heesterbeek (2003) A new method for estimating the effort required to control an infectious disease, Pro. Roy. Soc. London B 270: 1359–1364. 37. Scalia-Tomba, G., A. Svensson, T. Asikaiainen, and J. Giesecke (2010) Some model based considerations on observing generation times for communicable diseases, Math. Biosc. 223: 24–31. 38. Shuai, Z. & P. van den Driessche (2011) Global dynamics of cholera models with differential infectivity, Math. Biosc. 234: 118–126. 39. Strogatz, S.H. (2001) Exploring complex networks, Nature, 410: 268–276. 40. Svensson, A. (2007) A note on generation time in epidemic models, Math. Biosc. 208: 300– 311. 41. Tien, J.H. & D.J.D. Earn (2010) Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol. 72: 1506–1533. 42. van den Driessche, P. & J. Watmough (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosc., 180: 29– 48. 43. Volz, E. (2008) SIR dynamics in random networks with heterogeneous connectivity, J. Math. Biol., 56: 293–310. 44. Wallinga, J. & M. Lipsitch (2007) How generation intervals shape the relationship between growth rates and reproductive numbers, Proc. Royal Soc. B 274: 599–604. 45. Wallinga, J. & P. Teunis (2004) Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures, Am. J. Epidem. 160: 509–516, 46. Wearing, H.J., P. Rohani & M. J. Keeling (2005) Appropriate models for the management of infectious diseases, PLOS Medicine, 2: 621–627. 47. Yan, P. and Z. Feng (2010) Variability order of the latent and the infectious periods in a deterministic SEIR epidemic model and evaluation of control effectiveness Math. Biosc. 224: 43–52. 48. Yang, C.K. and F. Brauer (2009) Calculation of R0 for age-of-infection models, Math. Biosc. & Eng. 5: 585–599

Chapter 5

Models with Heterogeneous Mixing

5.1 A Vaccination Model To cope with annual seasonal influenza epidemics there is a program of vaccination before the “flu” season begins. Each year a vaccine is produced aimed at protecting against the three influenza strains considered most dangerous for the coming season. We formulate a model to add vaccination to the simple SI R model under the assumption that vaccination reduces susceptibility (the probability of infection if a contact with an infected member of the population is made). We consider a population of total size N and assume that a fraction γ of this population is vaccinated prior to a disease outbreak. Thus we have unvaccinated and vaccinated sub-populations of sizes NU = (1 − γ )N and NV = γ N, respectively. We assume that vaccinated members have susceptibility to infection reduced by a factor σ, 0 ≤ σ ≤ 1, with σ = 0 describing a perfectly effective vaccine and σ = 1 describing a vaccine that has no effect. We assume also that vaccinated individuals who are infected have infectivity reduced by a factor δ and both vaccinated and unvaccinated individuals have a recovery rate α. The number of contacts in unit time per member is aU for unvaccinated individuals and aV for vaccinated individuals. These may be equal. In this chapter we will study models in which there is more than one susceptible or infective compartment, and it is convenient to formulate such models using the number of contacts in unit time instead of the number of contacts multiplied by the total population size. Thus, for example, instead of writing the simple epidemic model as S ′ = βSI,

I ′ = βSI − αI,

© Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_5

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5 Models with Heterogeneous Mixing

we would write it as S ′ = −aS

I , N

I ′ = aS

I − αI. N

We let SU , SV , IU , IV denote the number of unvaccinated susceptibles, the number of vaccinated susceptibles, the number of unvaccinated infectives, and the number of vaccinated infectives, respectively. The resulting model is 

 IU IV = −aU SU +δ NU NV   IU IV SV′ = −σ aV SV +δ NU NV   IU IV IU′ = aU SU − αIU +δ NU NV   IU IV ′ − αIV . IV = σ aV SV +δ NU NV

SU′

(5.1)

The initial conditions prescribe SU (0), SV (0), IU (0), IV (0), with SU (0) + IU (0) = NU ,

SV (0) + IV (0) = NV .

Since the infection now is beginning in a population which is not fully susceptible, we speak of the control reproduction number Rc rather than the basic reproduction number. However, as we will soon see, calculation of the control reproduction number will require a more general definition and a considerable amount of technical computation. The computation method is applicable to both basic and control reproduction numbers. We will use the term reproduction number to denote either a basic reproduction number or a control reproduction number. We are able to obtain final size relations without knowledge of the reproduction number but these final size relations do contain information about the reproduction number, and more. Since SU and SV are decreasing non-negative functions they have limits SU (∞) and SV (∞), respectively, as t → ∞. The sum of the equations for SU and IU in (5.1) is (SU + IU )′ = −αIU , from which we conclude, just as in the analysis of the simple SI R model in Sect. 2.4, that IU (t) → 0 as t → ∞, and that α



∞ 0

IU (t)dt = NU − SU (∞).

(5.2)

5.1 A Vaccination Model

181

Similarly, using the sum of the equations for SV and IV , we see that IV (t) → 0 as t → ∞, and that α



∞

0

IV (t)dt = NV − SV (∞).

(5.3)

Integration of the equation for SU in (5.1) and use of (5.2), (5.3) gives SU (0) = aU log SU (∞)



0

∞

IU (t)dt + δ



∞

IV (t)dt

0



    SU (∞) SV (∞) δaU aU + . 1− 1− = α NU α NV

(5.4)

A similar calculation using the equation for SV gives σ aV SV (0) = log SV (∞) α

    SU (∞) SV (∞) δσ aV 1− 1− + . NU α NV

(5.5)

This pair of Eqs. (5.4) and (5.5) are the final size relations. They make it possible to calculate SU (∞), SV (∞) if the parameters of the model are known. It is convenient to define the matrix    aU δaU  R11 R12 R= = σ αaV δσαaV . R21 R22 αU αV The element Rij can be interpreted as the average number of susceptibles of group i infected by an infective of type j over its infectious period. Then the final size relations (5.4), (5.5) may be written as     SV (∞) SU (∞) SU (0) + R12 1 − = R11 1 − SU (∞) NU NV     SV (∞) SU (∞) SV (0) + R22 1 − . = R21 1 − log SV (∞) NU NV

log

(5.6)

The matrix R is closely related to the reproduction number. In the next section we describe a general method for calculating reproduction numbers that will involve a matrix which is similar to this matrix.

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5 Models with Heterogeneous Mixing

5.2 The Next Generation Matrix and the Basic Reproduction Number Up to this point, we have calculated reproduction numbers by following the secondary cases caused by a single infective introduced into a population. However, if there are sub-populations with different susceptibility to infection, as in the vaccination model introduced in Sect. 5.1, it is necessary to follow the secondary infections in the sub-populations separately, and this approach will not yield the reproduction number. It is necessary to give a more general approach to the meaning of the reproduction number, and this is done through the next generation matrix [18, 19, 45]. The underlying idea is that we must calculate the matrix whose (i, j ) entry is the number of secondary infections caused in compartment i by an infected individual in compartment j . We will follow the development in [45, 46] for ordinary differential equation models, even though the approach of [18, 19] is more general. In a compartmental disease transmission model, we sort individuals into compartments based on a single, discrete state variable. A compartment is called a disease compartment if the individuals therein are infected. Note that this use of the term disease is broader than the clinical definition and includes stages of infection such as exposed stages in which infected individuals are not necessarily infective. Suppose there are n disease compartments and m non-disease compartments, and let x ∈ R n and y ∈ R m be the sub-populations in each of these compartments. Further, we denote by Fi the rate at which secondary infections increase the i th disease compartment and by Vi the rate at which disease progression, death, and recovery decrease the i th compartment, that is, Vi is the net outflow from the i th compartment due to disease progression, death, and recovery, with inflow from other compartments yielding a negative contribution. The compartmental model can then be written in the form xi′ = Fi (x, y) − Vi (x, y), i = 1, . . . , n, yj′ = gj (x, y), j = 1, . . . , m.

(5.7)

Note that the decomposition of the dynamics into F and V and the designation of compartments as infected or uninfected may not be unique; different decompositions correspond to different epidemiological interpretations of the model. The definitions of F and V used here differ slightly from those in [45]. The derivation of the basic reproduction number is based on the linearization of the ODE model about a disease-free equilibrium. We make the following assumptions: • Fi (0, y) = 0 and Vi (0, y) = 0 for all y ≥ 0 and i = 1, . . . , n. • The disease-free system y ′ = g(0, y) has a unique equilibrium that is asymptotically stable, that is, all solutions with initial conditions of the form (0, y) approach a point (0, yo ) as t → ∞. We refer to this point as the disease-free equilibrium.

5.2 The Next Generation Matrix and the Basic Reproduction Number

183

The first assumption says that all new infections are secondary infections arising from infected hosts; there is no immigration of individuals into the disease compartments. It ensures that the disease-free set, which consists of all points of the form (0, y), is invariant. That is, any solution with no infected individuals at some point in time will be free of infection for all time. The second assumption ensures that the disease-free equilibrium is also an equilibrium of the full system. Next, we further assume that • Fi (x, y) ≥ 0 for all non-negative x and y and i = 1, . . . , n. • V y) ≤ 0 whenever xi = 0, i = 1, . . . , n.

i (x, n • i=1 Vi (x, y) ≥ 0 for all non-negative x and y.

The reasons for these assumptions are that the function F represents new infections and cannot be negative, each component, Vi , represents a net outflow from compartment i and

must be negative (inflow only) whenever the compartment is empty, and the sum ni=1 Vi (x, y) represents the total outflow from all infected

compartments. Terms in the model leading to increases in ni=1 xi are assumed to represent secondary infections and therefore belong in F . Suppose that a single infected person is introduced into a population originally free of disease. The initial ability of the disease to spread through the population is determined by an examination of the linearization of (5.7) about the disease-free equilibrium (0, yo ). It is easy to see that the assumption Fi (0, y) = 0, Vi (0, y) = 0 implies ∂Vi ∂Fi (0, yo ) = (0, yo ) = 0 ∂yj ∂yj for every pair (i, j ). This implies that the linearized equations for the disease compartments, x, are decoupled from the remaining equations and can be written as x ′ = (F − V )x,

(5.8)

where F and V are the n × n matrices with entries F =

∂Fi (0, yo ) ∂xj

and

V =

∂Vi (0, yo ). ∂xj

Because of the assumption that the disease-free system y ′ = g(0, y) has a unique asymptotically stable equilibrium, the linear stability of the system (5.7) is completely determined by the linear stability of the matrix (F − V ) in (5.8). The number of secondary infections produced by a single infected individual can be expressed as the product of the expected duration of the infective period and the rate at which secondary infections occur. For the general model with n disease compartments, these are computed for each compartment for a hypothetical index case. The expected time the index case spends in each compartment is given by ∞ the integral 0 φ(t, x0 ) dt, where φ(t, x0 ) is the solution of (5.8) with F = 0

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5 Models with Heterogeneous Mixing

(no secondary infections) and non-negative initial conditions, x0 , representing an infected index case: x ′ = −V x,

x(0) = x0 .

(5.9)

In effect, this solution shows the path of the index case through the disease compartments from the initial exposure through death or recovery with the i th component of φ(t, x0 ) interpreted as the probability that the index case (introduced at time t = 0) is in disease state i at time t. The solution of (5.9) is φ(t, x0 ) = e−V t x0 , where the exponential of a matrix is defined by the Taylor series A3 Ak A2 + + ··· + + ··· . 2 3! k! ∞ This series converges for all t and 0 φ(t, x0 ) dt = V −1 x0 (see, for example, [29]). The (i, j ) entry of the matrix V −1 can be interpreted as the expected time an individual initially introduced into disease compartment j spends in disease compartment i. The (i, j ) entry of the matrix F is the rate at which secondary infections are produced in compartment i by an index case in compartment j . Hence, the expected number of secondary infections produced by the index case is given by eA = I + A +



0

∞

F e−V t x0 dt = F V −1 x0 .

Following Diekmann and Heesterbeek [18], the matrix KL = F V −1 is referred to as the next generation matrix with large domain for the system at the disease-free equilibrium. The (i, j ) entry of K is the expected number of secondary infections in compartment i produced by individuals initially in compartment j , assuming, of course, that the environment seen by the individual remains homogeneous for the duration of its infection. Shortly, we will describe some results from matrix theory which imply that the matrix, KL = F V −1 is non-negative and therefore has a non-negative eigenvalue, R0 = ρ(F V −1 ), such that there are no other eigenvalues of K with modulus greater than R0 and there is a non-negative eigenvector ω associated with R0 [7, Theorem 1.3.2]. This eigenvector is in a sense the distribution of infected individuals that produces the greatest number, R0 , of secondary infections per generation. Thus, R0 and the associated eigenvector ω suitably define a “typical” infective and the basic reproduction number can be rigorously defined as the spectral radius of the matrix, KL . The spectral radius of a matrix KL , denoted by ρ(KL ), is the maximum of the moduli of the eigenvalues of KL . If KL is irreducible, then R0 is a simple eigenvalue of KL and is strictly larger in modulus than all other eigenvalues of KL . However, if KL is reducible, which is often the case for diseases with

5.2 The Next Generation Matrix and the Basic Reproduction Number

185

multiple strains, then KL may have several positive real eigenvectors corresponding to reproduction numbers for each competing strain of the disease. We have interpreted the reproduction number for a disease as the number of secondary infections produced by an infected individual in a population of susceptible individuals. If the reproduction number R0 = ρ(F V −1 ) is consistent with the differential equation model, then it should follow that the disease-free equilibrium is asymptotically stable if R0 < 1 and unstable if R0 > 1. This is shown through a sequence of lemmas. The spectral bound (or abscissa) of a matrix A is the maximum real part of all eigenvalues of A. If each entry of a matrix T is non-negative we write T ≥ 0 and refer to T as a non-negative matrix. A matrix of the form A = sI − B, with B ≥ 0, is said to have the Z sign pattern. These are matrices whose off-diagonal entries are negative or zero. If in addition, s ≥ ρ(B), then A is called an M-matrix. Note that in this section, I denotes an identity matrix, not a population of infectious individuals. The following lemma is a standard result from [7]. Lemma 5.1 If A has the Z sign pattern, then A−1 ≥ 0 if and only if A is a nonsingular M-matrix. The assumptions we have made imply that each entry of F is non-negative and that the off-diagonal entries of V are negative or zero. Thus V has the Z sign pattern. Also, the column sums of V are positive or zero, which, together with the Z sign pattern, implies that V is a (possibly singular) M-matrix [7, condition M35 of Theorem 6.2.3]. In what follows, it is assumed that V is non-singular. In this case, V −1 ≥ 0, by Lemma 5.1. Hence, KL = F V −1 is also non-negative. Lemma 5.2 If F is non-negative and V is a non-singular M-matrix, then R0 = ρ(F V −1 ) < 1 if and only if all eigenvalues of (F − V ) have negative real parts. Proof Suppose F ≥ 0 and V is a non-singular M-matrix. By the proof of Lemma 5.1, V −1 ≥ 0. Thus, (I − F V −1 ) has the Z sign pattern, and by Lemma 5.1, (I − F V −1 )−1 ≥ 0 if and only if ρ(F V −1 ) < 1. From the equalities (V − F )−1 = V −1 (I − F V −1 )−1 and V (V − F )−1 = I + F (V − F )−1 , it follows that (V −F )−1 ≥ 0 if and only if (I −F V −1 )−1 ≥ 0. Finally, (V −F ) has the Z sign pattern, so by Lemma 5.1, (V − F )−1 ≥ 0 if and only if (V − F ) is a non-singular M-matrix. Since the eigenvalues of a non-singular M-matrix all have positive real parts, this completes the proof. ⊔ ⊓ Theorem 5.1 Consider the disease transmission model given by (5.7). The diseasefree equilibrium of (5.7) is locally asymptotically stable if R0 < 1, but unstable if R0 > 1.

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5 Models with Heterogeneous Mixing

Proof Let F and V be defined as above, and let J21 and J22 be the matrices of partial derivatives of g with respect to x and y evaluated at the disease-free equilibrium. The Jacobian matrix for the linearization of the system about the disease-free equilibrium has the block structure J =



 F −V 0 . J21 J22

The disease-free equilibrium is locally asymptotically stable if the eigenvalues of the Jacobian matrix all have negative real parts. Since the eigenvalues of J are those of (F − V ) and J22 , and the latter all have negative real parts by assumption, the disease-free equilibrium is locally asymptotically stable if all eigenvalues of (F −V ) have negative real parts. By the assumptions on F and V , F is non-negative and V is a non-singular M-matrix. Hence, by Lemma 5.2 all eigenvalues of (F − V ) have negative real parts if and only if ρ(F V −1 ) < 1. It follows that the disease-free equilibrium is locally asymptotically if R0 = ρ(F V −1 ) < 1. Instability for R0 > 1 can be established by a continuity argument. If R0 ≤ 1, then for any ǫ > 0, ((1 + ǫ)I − F V −1 ) is a non-singular M-matrix and, by Lemma 5.1, ((1 + ǫ)I − F V −1 )−1 ≥ 0. By the proof of Lemma 5.2, all eigenvalues of ((1 + ǫ)V − F ) have positive real parts. Since ǫ > 0 is arbitrary, and eigenvalues are continuous functions of the entries of the matrix, it follows that all eigenvalues of (V − F ) have non-negative real parts. To reverse the argument, suppose all the eigenvalues of (V − F ) have non-negative real parts. For any positive ǫ, (V + ǫI − F ) is a non-singular M-matrix, and by the proof of Lemma 5.2, ρ(F (V + ǫI )−1 ) < 1. Again, since ǫ > 0 is arbitrary, it follows that ρ(F V −1 ) ≤ 1. Thus, (F − V ) has at least one eigenvalue with positive real part if and only if ρ(F V −1 ) > 1, and the disease-free equilibrium is unstable whenever R0 > 1. ⊔ ⊓ These results validate the extension of the definition of the reproduction number to more general situations. In the vaccination model (5.1) of Sect. 5.1 we calculated a pair of final size relations which contained the elements of a matrix K. This matrix is precisely the next generation matrix with large domain KL = F V −1 that has been introduced in this section. Example 1 Consider the SEI R model with infectivity in the exposed stage, a S ′ = − S(I + εE) N a E ′ = S(I + εE) − κE N I ′ = κE − αI R ′ = αI.

(5.10)

5.2 The Next Generation Matrix and the Basic Reproduction Number

187

Here the disease states are E and I , F =

  εEa + I a 0

and 

 εa a F = , 0 0



 κ 0 V = , −κ α

V

−1



 1/κ 0 = . 1/α 1/α

Then we may calculate KL = F V −1 =

 εa

a a κ α α . 0 0 +

It is clear that since R0 is equal to the trace of F V −1 , R0 =

εa a + , κ α

the element in the first row and first column of F V −1 . If all new infections are in a single compartment, as is the case here, the basic reproduction number is the trace of the matrix F V −1 . In general, it is possible to reduce the size of the next generation matrix with large domain to the number of state at infection [18]. The states at infection are those disease states in which there can be new infections. Suppose that there are n disease states and k states at infection with k < n. Then we may define an auxiliary n × k matrix P in which each column corresponds to a state at infection and has 1 in the corresponding row and 0 elsewhere. Then the next generation matrix is the k × k matrix K = P T KL P . It is easy to show, using the fact that P P T KL = KL , that the n×n matrix KL and the k × k matrix K have the same non-zero eigenvalues and therefore the same spectral radius. Construction of the next generation matrix which has lower dimension than the next generation matrix with large domain may simplify the calculation of the basic reproduction number. In Example 1 above, the only disease state at infection is E, the matrix P is   1 , 0

188

5 Models with Heterogeneous Mixing

and the next generation matrix K is the 1 × 1 matrix K=

 εa κ

+

a . α

5.2.1 Some More Complicated Examples The next generation approach is very general and can be applied to models with heterogeneous mixing and control measures applied differently in different groups. Example 2 Consider the vaccination model (5.1) of Sect. 5.1. The disease states are IU and IV . Then F =



aU (IU + δIV ) σ aV (IU + δIV



and  aU δaU , F = σ aV σ δaV 

  αU 0 V = . 0 αV

It is easy to see that the next generation matrix with large domain is the matrix K calculated in Sect. 5.1. Since each disease state is a disease state at infection, the next generation matrix is K, the same as the next generation matrix with large domain. As in Example 1, the determinant of K is zero and K has rank 1. Thus the control reproduction number is the trace of K, Rc =

δσ NV wU + . αU αV

5.3 Heterogeneous Mixing In disease transmission models not all members of the population make contacts at the same rate. In sexually transmitted diseases there is often a “core” group of very active members who are responsible for most of the disease cases, and control measures aimed at this core group have been very effective in control [27]. In epidemics there are often “super-spreaders”, who make many contacts and are instrumental in spreading disease and in general some members of the population make more contacts than others. Recently there has been a move to complicated network models for simulating epidemics [23, 24, 32–34, 38]. These assume knowledge of the mixing patterns of groups of members of the population and

5.3 Heterogeneous Mixing

189

make predictions based on simulations of a stochastic model. A basic description of network models may be found in [44]. While network models can give very detailed predictions, they have some serious disadvantages. For a detailed network model, simulations take long enough to make it difficult to examine a significant range of parameter values, and it is difficult to estimate the sensitivity with respect to parameters of the model. The theoretical analysis of network models is a very active and rapidly developing field [38–40]. However, it is possible to consider models more realistic than simple compartmental models but simpler to analyze than detailed network models. To model heterogeneity in mixing we may assume that the population is divided into subgroups with different activity levels. We will analyze an SI R model in which there are two groups with different contact rates. The approach extends easily to models with more compartments, such as exposed periods or a sequence of infective stages and also to models with an arbitrary number of activity levels. In this way, we may hope to give models intermediate between the too simple compartmental models and the too complicated network models. In this section, we describe the formulation of models for two groups with different activity levels and give the main results for the simplest compartmental epidemic models. The analysis of models of the same type with more complicated compartmental structure is given in [11] and the analysis of models with more groups is given in [12]. There is no problem, other than technical calculation difficulties, in extending everything in this section to an arbitrary number of subpopulations. Consider two sub-populations of constant sizes N1 , N2 , respectively, each divided into susceptibles, infectives, and removed members with subscripts to identify the sub-population. In this section, we will assume that the number of contacts per member in unit time is a constant. Suppose that each group i member makes ai contacts sufficient to transmit infection in unit time, and that the fraction of contacts made by a member of group i that is with a member of group j is pij , (i, j = 1, 2). Then pi1 + pi2 = 1,

i = 1, 2.

We assume that all contacts between a susceptible and an infective transmit infection to the susceptible. Suppose the mean infective period in group i is 1/αi . We assume that there are no disease deaths, so that the population size of each group is constant. A two-group SI R epidemic model is   I1 I2 + pi2 Si′ = −ai Si pi1 N1 N2   I1 I2 − αi Ii , Ii′ = ai Si pi1 + pi2 N1 N2

(5.11) i = 1, 2.

190

5 Models with Heterogeneous Mixing

The initial conditions are Si (0) + Ii (0) = Ni ,

i = 1, 2.

The two-group model includes two possibilities. It may describe a population with groups differing by activity levels and possibly by vulnerability to infection. For an epidemic model, in which we assume the time scale is short enough that members do not age over the course of the epidemic, the grouping could be by age. However, for a longer term disease transmission model with age-dependent transmission it would be necessary to take into account the fact that the ages of members of the population change over the course of the disease and use a different type of model, to be studied in Chap. 13. The two-group model (5.11) assumes the same infectivities and susceptibilities in each group. A more general model would be   I1 I2 Si′ = −σi ai Si δ1 pi1 + δ2 pi2 N1 N2   I2 I1 − αi Ii , Ii′ = σi ai Si δ1 pi1 + δ2 pi2 N1 N2

(5.12) i = 1, 2.

This is just the model (5.11) with the addition of susceptibility factors σ1 , σ2 for susceptibles in the two groups and infectivity factors δ1 , δ2 for infectives in the two groups. As before a1 , a2 are effective contact rates, and this model adds transmission probabilities to (5.11). It is not possible to calculate the reproduction number for the two-group model (5.11) directly by counting secondary infections. It is necessary to use the next generation matrix approach of [45] described in Section 5.2 and calculate the reproduction number as the largest eigenvalue of the matrix F V −1 , where ⎡ N1 ⎤   p11 a1 p12 a1 α1 0 ⎢ N2 ⎥ F =⎣ , V = . ⎦ N2 0 α2 p22 a2 p21 a2 N1 Then

F V −1

⎡ p a p12 a1 N1 ⎤ 11 1 ⎢ α2 N2 ⎥ 1 = ⎣p α p22 a2 ⎦ . 21 a2 N2 α1 N1 α2

The eigenvalues of the matrix F V −1 are the roots of the quadratic equation 2

λ −



p11 a1 p22 a2 + α1 α2



λ + (p11 p22 − p12 p21 )

a1 a2 = 0. α1 α2

(5.13)

5.3 Heterogeneous Mixing

191

The basic reproduction number R0 is the larger of these two eigenvalues,

R0 =

p22 a2 p11 a1 + + α1 α2

"



p11 a1 p22 a2 − α1 α2 2

2

+4

p12 p21 a1 a2 α1 α2

.

In order to obtain a more useful expression for R0 , it is necessary to make some assumptions about the nature of the mixing between the two groups. The mixing is determined by the two quantities p12 , p21 since p11 = 1 − p12 and p22 = 1 − p21 . There has been much study of mixing patterns, see, for example, [8, 9, 13]. One possibility is proportionate mixing, that is, that the number of contacts between groups is proportional to the relative activity levels. In other words, mixing is random but constrained by the activity levels [42]. Under the assumption of proportionate mixing, pij =

aj Nj , a1 N1 + a2 N2

and we may write p11 = p21 = p1 ,

p12 = p22 = p2 ,

with p1 + p2 = 1. In particular, p11 p22 − p12 p21 = 0, and thus R0 = a1

p2 p1 + a2 . α1 α2

Another possibility is preferred mixing [42], in which a fraction πi of each group mixes randomly with its own group and the remaining members mix proportionately. Thus, preferred mixing is given by p11 = π1 + (1 − π1 )p1 , p12 = (1 − π1 )p2 p21 = (1 − π2 )p1 , p22 = π2 + (1 − π2 )p2 ,

(5.14)

with pi =

(1 − πi )ai Ni , (1 − π1 )a1 N1 + (1 − π2 )a2 N2

i = 1, 2.

Proportionate mixing is the special case of preferred mixing with π1 = π2 = 0.

192

5 Models with Heterogeneous Mixing

It is also possible to have like-with-like mixing, in which members of each group mix only with members of the same group. This is the special case of preferred mixing with π1 = π2 = 1. For like-with-like mixing, p11 = p22 = 1,

p12 = p21 = 0.

Then the roots of (5.13) are a1 /α1 and a2 /α2 , and the reproduction number is R0 = max

#a

1

,

α1

a2 $ . α2

By calculating the partial derivatives of pij (i, j = 1, 2) with respect to π1 and π2 , we may show that p11 and p22 increase when either π1 or π2 is increased, while p12 and p21 decrease when either π1 or π2 is increased. From this, we may see from the general expression for R0 that increasing either of the preferences π1 , π2 increases the basic reproduction number. We may follow the analysis of the SI R model (5.11) to obtain the basic reproduction number for the SI R model (5.12) with susceptibility and infectivity reduction factors "   2 p12 p21 a1 a2 p22 a2 2 p11 a1 pii ai σ1 δ1 + 4σ1 δ2 σ2 δ1 + − σ2 δ2 σi δi αi α1 α2 α1 α2 i=1 R0 = . 2 In the special case of proportionate mixing, where p11 p22 −p12 p21 = 0, this reduces to R0 =

2 i=1

σi δi

pii ai . αi

The vaccination model of Sect. 5.1 is an example of a two-group model of the form (5.12), with σ1 = σ2 = δ1 = 1,

δ2 = δ.

It is easy to show [11] that, just as for a one-group model [10], S1 → S1 (∞) > 0, as t → ∞.

S2 → S2 (∞) > 0,

5.3 Heterogeneous Mixing

193

Example 1 Consider a two-group SEI R model,  I Si′ = −ai Si pii NIii + pij Njj  I Ei′ = ai Si pii NIii + pij Njj − κi Ei

(5.15)

Ii′ = κi Ei − αi Ii Ri′ = αi Ii , i, j = 1, 2, i = j.

The disease states are Ei and Ii (i = 1, 2). Now ⎡ N1 ⎤ a1 p11 I1 + a1 p12 I2 ⎢ N2 ⎥ ⎢ ⎥ 0 ⎢ ⎥ F =⎢ ⎥, N2 ⎢ ⎥ a2 p21 I1 ⎣ ⎦ N1 0

⎡

0

a1 p11

N1 ⎤ N2 ⎥ ⎥ 0 ⎥ ⎥, a2 p22 ⎥ ⎦

0 a1 p12

⎢ ⎢ 0 0 ⎢0 F =⎢ N 2 ⎢0 a p 0 2 21 ⎣ N1 0 0 0

0

and ⎡

κ1 ⎢−κ1 V =⎢ ⎣ 0 0

0 α1 0κ2 0

⎤ 0 0 0 0⎥ ⎥, ⎦ 0 −κ2 α2

⎡

a1

V −1

⎡

⎤ 0 0 0 0⎥ ⎥ =⎢ ⎥. ⎣ 0 0 κ12 0 ⎦ 0 0 α12 α12 1 κ1 0 ⎢1 1 ⎢ α1 α1

Then we may calculate

KL = F V −1

p11 p12 N1 p12 N1 ⎤ p11 a1 a1 a1 ⎢ α1 α1 α1 N2 α1 N2 ⎥ ⎢ ⎥ 0 0 0 0 ⎢ ⎥ . =⎢ p N p22 ⎥ ⎥ ⎢a 21 2 a p21 N2 a p22 a2 2 2 ⎦ ⎣ 2 α1 N1 α1 N1 α2 α2 0 0 0 0

In this example, it is advantageous to construct the next generation matrix by reducing the next generation matrix with large domain KL . In order to do this, we use the auxiliary matrix ⎡ 1 ⎢0 E=⎢ ⎣0 0

⎤ 0 0⎥ ⎥ 1⎦ 0

194

5 Models with Heterogeneous Mixing

to construct the next generation matrix ⎡ p a p12 a1 N1 ⎤ 11 1 ⎢ α2 N2 ⎥ 1 K = E T KL E = ⎣ p α p22 a2 ⎦ . 21 a2 N2 α1 N1 α2

(5.16)

This is the same matrix as the next generation matrix obtained for the two-group SI R model (5.11) introduced in this section. For a one-group epidemic model there is a final size relation that makes it possible to calculate the size of the epidemic from the reproduction number [10, 35]. There is a corresponding final size relation for the two-group model (5.1), established in much the same way, combining expressions for the integrals of (S1 + I1 )′ , (S2 + I2 )′ , (S1 )′ /S1 , (S2 )′ /S2 ). This relation does not involve the reproduction number explicitly but still makes it possible to calculate the size of the epidemic from the model parameters. The final size relation for the model (5.11) is the pair of equations      pij Sj (∞) Si (∞) pii Si (0) 1− + 1− , i, j = 1, 2, i = j. = ai log Si (∞) αi Ni αj Nj (5.17) Just as with the vaccination model (5.1), the final size relation may be written in terms of the matrix    a1 p11 a1 p12  R11 R12 R= = a2αp121 a2αp222 . R21 R22 α1 α2 The matrix R is similar to the next generation matrix K (and therefore has the same eigenvalues), since R = T −1 KT , where T =

  N1 0 . 0 N2

The final size relation makes it possible to calculate S1 (∞) and S2 (∞) and thence the number of disease cases [N1 − S1 (∞)] + [N2 − S2 (∞)].

5.3 Heterogeneous Mixing

195

The final size relation takes a simpler form if the mixing is proportionate. With proportionate mixing, since p11 = p21 = p1 ,

p12 = p22 = p2 ,

(5.17) implies a2 log

S1 (0) S2 (0) = a1 log , S1 (∞) S2 (∞)

and we may write the final size relations as     S1 (∞) a1 p2 S2 (∞) a1 p1 S1 (0) 1− + 1− , = log S1 (∞) α1 N1 α2 N2     S1 (∞) a2 S2 (∞) a1 = . S1 (0) S2 (0)

(5.18)

We recall that in the case of proportionate mixing R0 =

p2 a2 p1 a1 + . α1 α2

The second equation of (5.18) implies that if a1 > a2 , then 1−

S2 (∞) S1 (∞) >1− , S1 (0) S2 (0)

that is, the attack ratio is greater in the more active group. It is not difficult to show that the final size relations (5.18) give a unique set of final numbers of susceptibles in each group. The final size relation can also be obtained in a similar way for more complicated compartmental models [4–6]. The model can be extended easily to an arbitrary number of groups with different activity levels. It is also important to be able to describe models with more stages in the progression through compartments, and models in which there are differences between groups in susceptibility. For example, influenza has two characteristics not included in the model (5.11) that are of importance. There is a latent period between infection and the development of infectivity and influenza symptoms. Also, only a fraction of latent members develop symptoms, while the remainder go through an asymptomatic stage in which there is some infectivity. Another important aspect is treatment, which could be directed at either or both group, and could be used for making decisions on how to target groups for treatment. A natural way to proceed in this direction is to build an age of infection model for populations with multiple groups.

196

5 Models with Heterogeneous Mixing

Example 2 Consider a two-group endemic SI R model with preferential mixing and group-targeted vaccination: dSi = μNi (1 − φi ) − (λi (t) + μ)Si , dt dIi = λi (t)Si − (α + μ)Ii , dt dRi = μNi φi + αIi − μRi , i = 1, 2, dt

(5.19)

where Ni = Si + Ii + Ri . Here, λi represents the force of infection for susceptibles in group i given by λi = ai σ

n

pij

j =1

Ij , Nj

(5.20)

where ai denotes the average number of contacts an individual in sub-population i has per unit of time (which represents the activity level of group i), and σ is the probability of infection per contact when the contact is with an infectious individual, φi denotes the proportion of susceptibles in group i vaccinated (and removed) when entering the population. The fraction Ij /Nj gives the probability that a contact is with an infectious individual in sub-population j . The contact matrix (pij ) has the same form as the preferential mixing considered earlier with pij = πi δij + (1 − πi )pj ,

i, j = 1, 2.

(5.21)

The parameter πi is the fraction of contacts with individuals in the same subpopulation, δij is the Kronecker delta (i.e., 1 when i = j and 0 otherwise), and pj =

(1 − πj )aj Nj , (1 − π1 )a1 N1 + (1 − π2 )a2 N2

j = 1, 2.

Clearly, unless all the sub-groups are isolated (i.e., no interactions between the groups), there must be some i with πi < 1. For each sub-population i, if all contacts are with people within the same group (i.e., pii = 1 and pij = 0 for i = j ), then the basic and control reproduction numbers for group i are, respectively, R0i =

σ ai , Rvi = R0i (1 − φi ), μ+α

i = 1, 2.

(5.22)

When there are contacts between sub-populations, i.e., pii < 1 or πi < 1 for some i, we can derive the basic and control reproduction numbers for the metapopulation.

5.3 Heterogeneous Mixing

197

These reproduction numbers will be functions of R0i or Rvi . The next generation matrix Kv (v for vaccination) is Kv =

  Rv1 p11 Rv1 p12 . Rv2 p21 Rv2 p22

(5.23)

The control reproduction number Rv for the metapopulation is given by Rv =

 1 A + D + (A − D)2 + 4BC , 2

(5.24)

where A = R01 p11 (1 − φ1 ), B = R01 p12 (1 − φ1 ), C = R02 p21 (1 − φ2 ), D = R02 p22 (1 − φ2 ), and R0i (i = 1, 2) are given in (5.22). If φ1 = φ2 = 0, then Rv reduces to R0 =

1 R01 p11 + R02 p22 + 2



(R01 p11 − R02 p22 )2 + 4R01 p12 R02 p21 .

To study the effects of vaccination strategies, assume that R0 > 1 in the absence of vaccination and R01 > 1,

R02 > 1.

(5.25)

Let Ω = {(φ1 , φ2 )| 0 ≤ φ1 < 1, 0 ≤ φ2 < 1}.

(5.26)

Then each point (φ1 , φ2 ) ∈ Ω represents a vaccination strategy. Because we are interested in the case when the two groups are not isolated, either π1 < 1 or π2 < 1. This will be assumed for the results below. It can be shown that, for each fixed (φ1 , φ2 ) ∈ Ω, Rv increases with both π1 and π2 , i.e., ∂Rv > 0, ∂π1

∂Rv >0 ∂π2

for all (π1 , π2 ) ∈ Ω.

(5.27)

For each fixed (π1 , π2 ), there are different combinations of φ1 and φ2 that can reduce Rv to be below 1. For ease of presentation, consider the simpler case in which π1 = π2 = π,

198

5 Models with Heterogeneous Mixing

and consider Rv = Rv (π ) as a function of π . Then, for each fixed π ∈ [0, 1), the curve determined by Rv (π ) = 1 divides the region Ω into two parts: one is the region Ωπ = {(φ1 , φ2 )| 0 ≤ Rv (π ) < 1, (φ1 , φ2 ) ∈ Ω, 0 ≤ π < 1}, which includes all points above the curve (see Fig. 5.1), and another is the region Dπ = {(φ1 , φ2 )| Rv (π ) > 1, (φ1 , φ2 ) ∈ Ω, 0 ≤ π < 1}, which includes all points below the curve. It can be shown that the region Ωπ decreases as π increases and reduces to the region Ω ∗ as π → 0, while the region Dπ decreases as π decreases and reduces to the region D ∗ as π → 1 (see Fig. 5.1). All these curves intersect at a single point (φ1c , φ2c ) with φ1c = 1 −

1 , R01

φ2c = 1 −

1 . R02

(5.28)

We observe from Fig. 5.1 that the region Ω ∗ (lighter-shaded) is determined by the two inequalities φ1c < φ1 < 1,

φ2c < φ2 < 1,

(5.29)

where φ1c and φ2c are defined in (5.28). For region D ∗ (darker-shaded), the upper bound is determined by the line φ2 = −A φ1 + B, Fig. 5.1 Plot showing the regions Ω ∗ and D ∗ . Several curves of Rv (π ) = 1 for different π values are also shown, with the dashed curves corresponding to 0 < π < 1, the thin solid lines (boundary of Ω ∗ ) corresponding to π = 1, and the thick line corresponding to π = 0 (the upper bound of the region D ∗ ). The arrows indicate the direction of change of the curve Rv (π ) = 1 as π increases from 0 to 1. All of the Rv (π ) = 1 curves intersect at the single point (φ1c , φ2c ). Source: [15]

(5.30)

5.3 Heterogeneous Mixing

199

where A =

R01 a1 N1 , R02 a2 N2

(R01 − 1)a1 N1 + (R02 − 1)a2 N2 B= . R02 a2 N2

(5.31)

The two regions intersect at the point (φ1c , φ2c ). This result suggests that there is a “lower bound” for vaccination efforts (φ1 , φ2 ), above which the infection can be eradicated regardless of mixing patterns. Similarly, it provides an “upper bound” for vaccination efforts (φ1 , φ2 ), below which the infection cannot be eradicated regardless of mixing patterns (see the definitions for φ1∗ and φ2∗ defined in (5.32) and see Fig. 5.2 for an illustration of the lower and upper bound). For an “intermediate level” vaccination strategy (φ1 , φ2 ), mixing parameters π1 and π2 can play an important role in influencing the effect of vaccination strategies on reducing Rv . Thus, when designing vaccination strategies, one should take into consideration mixing patterns within and between sub-populations. v For given πi (i = 1, 2), it can be shown that ∂R ∂φi < 0. When the curve Rv = 1 lies between regions D ∗ and Ω ∗ , the curve intersects the φ1 -axis and φ2 -axis at (φ1∗ , 0) and (0, φ2∗ ), respectively, where 1 − R02 p22 , R01 p11 (1 − R02 p22 ) + R01 R02 p12 p21 1 − R01 p11 φ2∗ = 1 − . R02 p22 (1 − R01 p11 ) + R22 R01 p12 p21

φ1∗ = 1 −

(5.32)

Because R0i > 1 for i = 1, 2, it is possible that R01 p11 > 1 and/or R02 p22 > 1. Thus, it is possible that φ1∗ > 1 and/or φ2∗ > 1. When φ1∗ > 1, we know from ∂Rv /∂φ1 < 1 that Rv > 1 for any vaccination strategy (φ1 , 0). Thus, it is impossible to eradicate the infection if only sub-population 1 is vaccinated. The results described above are based on the control reproduction number. Figure 5.2 shows some simulation results illustrating the effect of vaccination on the prevalence of infection. Different preference levels are used: π1 = 0.2 and π2 = 0.4, i.e., group 2 has a higher preference contacting people in its own group. Other parameter values used are σ = 0.03, α = 0.15 (an infective period of about 6 days), and a1 = 12, a2 = 8, μ = 0.00016 (a duration of 17 years in school). These values correspond to R01 = 2.4 and R02 = 1.6. The initial conditions are x1 (0) = S1 (0)/N1 (0) = 0.4, y1 (0) = I1 (0)/N1 (0) = 0.00002, x2 (0) = S2 (0)/N2 (0) = 0.6, y2 (0) = I2 (0)/N2 (0) = 0.00002. For this set of parameters, φ1∗ = 0.77 and φ2∗ ≫ 1. Figure 5.2a is for a vaccination strategy (φ1 , 0) with φ1 = 0.2 < φ1∗ , for which the infection persists (Rv = 1.8 > 1), while Fig. 5.2b is for a vaccination strategy (φ1 , 0) with φ1 = 0.8 > φ1∗ , in which case the infection dies out (Rv = 0.97 < 1).

200

5 Models with Heterogeneous Mixing

Fig. 5.2 When φ1∗ < 1 and φ2∗ > 1, the disease is eventually eradicated if the vaccination is applied to sub-population 1 alone at a level above φ1∗ . (a) (φ1 , φ2 ) = (0.2, 0) and φ1 < φ1∗ = 0.77, the disease persists (Rv = 1.8); (b) (φ1 , φ2 ) = (0.8, 0) and φ1 > φ1∗ = 0.77, the disease eventually disappears (Rv = 0.97). Source: [15]

Age

60

40

20

20

40 Age

60

Fig. 5.3 The figure on the left shows observed patterns of contacts between age groups [41]. The lighter areas correspond to higher level of contacts. The figure on the right shows schematic contact matrices illustrating the main and off-diagonals representing contacts among contemporaries, between children and parents, and vice versa

Example 3 This example considers a more general preferential mixing function than the one given in (5.21). This is motivated by the observed mixing pattern shown in Fig. 5.3. Figure 5.3 illustrates the recently collected data reported in [41], which reveals the preferential mixing between parents and children in addition to that among contemporaries. In the case of n groups, the similar function as (5.21) can be written as pij = πi δij + (1 − πi )pj ,

i, j = 1, 2, · · · , n, j = i

(5.33)

5.3 Heterogeneous Mixing

201

with (1 − πi )ai Ni pi = n , k=1 (1 − πk )ak Nk

i = 1, 2, · · · , n.

Here, ai ’s are the contact rates, δij ’s are the Kronecker delta (i.e., δij = 1 if i = j and δij = 0 if i = j ). To capture the feature shown in Fig. 5.3 (left), Glasser et al. in [25] extended the mixing function (5.33) to include not only the preference along the diagonal but also along the sub- and super-diagonals as indicated in Fig. 5.3 (right). The function is described by 

pij = φij + 1 −

3



πli pj ,

l=1

with φij =

%





1 − 3l=1 πli aj Nj   pj =

3 n 1 − ak Nk π lk k=1 l=1

δij π1i + δi(j +G) π2i , δij π1i + δi(j −G) π3i ,

i ≥ G, i ≤ L − G.

(5.34)

(5.35)

G is the generation time (i.e., average age at which women bear children), L is longevity (i.e., average expectation of life at birth), and L > G. The parameters ε1i – ε3i represent the fractions of contacts reserved for contemporaries, children (j −G), and parents (j + G), respectively, and the corresponding delta function is defined as δi(j ±G) =

%

1 if i = j ± G, 0 otherwise.

(5.36)

Only people whose ages equal or exceed G can have children, and only those whose ages equal to or less than L − G can have parents, but people aged at least G but not more than L − G can have both children and parents. Denote the preferential vectors by Πl = (πl1 , πl2 , · · · , πln ), l = 1, 2, 3. When Π2 = Π3 = 0, the expression (5.34) reduces to the formula (5.33). For ease of notation, we mix indices and real numbers, but if age classes are 0–4, 5–9, · · · and G = 25 years, for example, by i > G we mean i > class 5. Notice that the non-zero elements of Π2 and Π3 are related. If G = 25 years, for example, then ai Ni π2i = aj Nj π3j , Notice also that 0 ≤

3

l=1 πli

< 1.

i = 6, 7, · · · , j = i − 5.

202

5 Models with Heterogeneous Mixing

The mixing function was further extended in [25] to replace the delta functions in (5.36) by Gaussian kernels. Delta formulations are convenient mathematically, but do not allow the age range of one’s contemporaries to vary as one ages (e.g., the range narrows perceptively among adolescents), much less differences between the age ranges of one’s contemporaries and one’s parents or children. By virtue of secular patterns in childbearing, moreover, the age ranges of parents and children may change with age. But the Gaussian formulation allows such variation, reproducing the essential features of these observations. Figure 5.4 demonstrates the comparison between observed mixing patterns (left panel) and the model outcomes (right panel) using the extended mixing functions.

5.3.1 *Optimal Vaccine Allocation in Heterogeneous Populations One of the significant benefits of models with heterogeneous mixing is that it provides an approach to identifying optimal allocation of vaccines in a metapopulation, particularly when resources are limited. Consider a metapopulation with n sub-populations connected by heterogeneous mixing, which is described by an n × n matrix P = (pij ) for i, j = 1, 2, · · · , n. The following model is studied in [21, 26, 43]: dSi = (1 − φi )θ Ni − (λi + θ )Si dt dIi = λi Si − (γ + θ )Ii dt dRi = φi θ Ni + γ Ii − θ Ri dt

(5.37)

Ni = Si + Ii + Ri λi = σ ai

n j =1

pij Ij /Nj ,

i = 1, 2, . . . , n,

where φi are proportions immunized at entry into sub-population i, γ is the percapita recovery rate, θ is the per-capita rate for entering and leaving sub-population i so that the population size Ni remains constant. The function λi is the force of infection, i.e., per-capita hazard rate of infection of susceptible individuals in sub-population i, in which σ is the probability of infection upon contacting an infectious person, ai is the average contact rate (activity) in sub-population i, pij is the proportion of ith sub-population’s contacts that are with members of j th subpopulation, and Ij /Nj is the probability that a randomly encountered member of sub-population j is infectious. A similar model with two-level mixing is studied in [22].

5.3 Heterogeneous Mixing

203

Fig. 5.4 Comparison of the mixing patterns generated by the four empirical studies (top to bottom): [17, 41, 47, 49] (left panel) and the fitted model (5.34) and (5.35) (right panel). Interpolating functions are fitted to geometric means of corresponding row- and column-elements of the mixing matrix Source: [25]

204

5 Models with Heterogeneous Mixing

The mixing matrix P can incorporate heterogeneities in contact rate (ai ), population size (Ni ), preference for mixing within the same sub-population (πi ), etc. Because of these heterogeneities, the optimal combination of vaccination coverages φi is unlikely to be homogeneous. Typically, the matrix P has to satisfy the following conditions of [14]: pij ≥ 0, i, j = 1, . . . , n, n pij = 1, i = 1, . . . , n, j =1

ai Ni pij = aj Nj pj i ,

(5.38)

i, j = 1, . . . , n.

A commonly used non-homogeneous mixing function that satisfies conditions in (5.38) is the preferred mixing function of [30] given by (1 − πj )aj Nj , pij = πi δij + (1 − πi ) n k=1 (1 − πk )ak Nk

i, j = 1, . . . , n,

(5.39)

where πi ∈ [0, 1] is the fraction of contacts of group i that is reserved for itself (preferential mixing), whereas the complement (1 − πi ) is distributed among all sub-populations in proportion to the unreserved contacts, including i (proportionate mixing). We refer to the mixing given by (5.39) as Jacquez-type preferred mixing. Two extreme cases of (5.39) are the proportionate mixing when πi = 0 and the isolated mixing (i.e., no interactions between sub-populations) when πi = 1. For model (5.37), the basic and effective sub-population reproduction numbers, denoted, respectively, by R0i and Rvi , for sub-population i (i = 1, 2, . . . , n) are given by R0i = ρai ,

Rvi = R0i (1 − φi ),

i = 1, 2, . . . , n,

(5.40)

where ρ=

σ . γ +θ

The next generation matrix (NGM) corresponding to this metapopulation model is ⎞ Rv1 p11 Rv1 p12 · · · Rv1 p1n ⎜ Rv2 p21 Rv2 p22 · · · Rv2 p2n ⎟ ⎟ ⎜ Kv = ⎜ . .. ⎟ . .. ⎝ .. . . ⎠ Rvn pn1 Rvn pn2 · · · Rvn pnn ⎛

Then the effective reproduction number for the metapopulation is given as Rv = r(Kv ),

(5.41)

5.3 Heterogeneous Mixing

205

which is the spectral radius (and the dominant eigenvalue, by the Perron–Frobenius Theorem) of the non-negative matrix Kv . Let φ = (φ1 , φ2 , · · · , φn ). Naturally, Rv = Rv (φ)

n is a function of φ. The total number of vaccine doses, denoted by η, is η = i=1 φi Ni . For demonstration purposes, we will assume that vaccine efficacy is 100%. We focus on identifying the most efficient allocation of vaccine φ = (φ1 , φ2 , · · · , φn ) ∈ [0, 1]n for reducing Rv with limited vaccine doses η or using fewest doses to achieve Rv < 1 (to prevent outbreaks). More specifically, we consider the following two constrained optimization problems: (I) (II)

Minimize Rv = Rv (φ), subject to ℓ(φ) := Minimize η =

n i=1

n i=1

φi Ni = η, for φ ∈ [0, 1]n .

φi Ni , subject to Rv (φ) ≤ 1, for φ ∈ [0, 1]n .

Consider the optimization problem only for the case of R0 = Rv (0) ≥ 1, as there will be no outbreak if R0 < 1. If a solution to Problem (I) exists for a given value of η, let Φ ∗ = Φ ∗ (η) and Rv{min} (η) denote the optimal vaccination allocation and corresponding minimum reproduction number, , respectively. Let # $ (n) Ωφ (η) := (φ1 , φ2 , · · · , φn ) : ℓ(φ) = η .

Then, for the solution Φ ∗ to be feasible, we need to have

Φ ∗ (η) = (φ1∗ (η), φ2∗ (η), · · · , φn∗ (η)) ∈ [0, 1]n , min Rv = Rv Φ ∗ (η) . Rv{min} (η) =

(5.42)

(n)

Ωφ (η)∩[0,1]n

An optimal solution Φ ∗ (η) to Problem (I) that lies in the interior of the unit hypercube must satisfy the following equations:  λ∇ℓ =  λ(N1 , · · · , Nn ), ∇Rv Φ ∗ (η) =  n  φi∗ (η)Ni = η, ℓΦ ∗ (η) =

Φ ∗ (η) ∈ (0, 1)n , (5.43)

i=1

where the constant  λ is the Lagrange multiplier. Similarly, if an optimal solution to Problem (II) exists, it is useful practically to have an explicit expression or estimate of the bounds for the minimum vaccine doses needed, which we denote by η∗ . The significance of η∗ is that it is the smallest number of vaccination doses that can prevent outbreaks under an optimal vaccination policy.

206

5 Models with Heterogeneous Mixing

To find η∗ , notice that Rv (φ) is a monotonically decreasing function of φi , and thus, a decreasing function of η = ni=1 φi Ni . Therefore, the inequality constraint Rv (φ) ≤ 1 can be replaced by an equality constraint Rv (φ) = 1, and thus, η∗ =

min

{Rv (φ)=1}∩[0,1]n

ℓ(φ).

It follows that η∗ is the minimum of η ∈ [0, N] such that Rv{min} (η) = 1 and can be found by solving the equation:  Rv{min} (η∗ ) = Rv P ∗ (η ) = 1. ∗

(5.44)

Before we present the results of optimal solutions for Problems (I) and (II), we state the following important results, which is proved in [43], regarding the bounds for the effective reproduction number Rv for general mixing matrices (not just Jacquez-type preferred mixing given in (5.39)). Theorem 5.2 (Bounds for Rv (φ)) Let P be a non-negative, invertible, irreducible matrix such that −P −1 is essentially non-negative and the conditions (5.38) are satisfied. Then (a) The lower and upper bounds of Rv (φ) are n i=1

ωi Rvi ≤ Rv ≤ max{Rv1 , . . . , Rvn },

ai Ni . where ωi = n k=1 ak Nk

(5.45)

(b) The lower and upper bounds of Rv (φ) correspond to the cases of proportionate mixing and isolated mixing, respectively. For the optimal solutions of Problems (I) and (II), it is shown in [43] that for the case of n = 2 explicit expressions for Φ ∗ and η∗ are possible, and for the case of n > 2 their upper and lower bounds can be obtained. For ease of presentation, introduce the following notation:   κ1 := p22 N1 N2 R02 − N2 p12 p21 R01 R02 ,   κ2 := p11 N1 N2 R01 − N1 p12 p21 R01 R02 ,

(5.46)

κ1 N1 + κ2 N2 . η0 := N − max{κ1 , κ2 }

For the mixing given in (5.39), it is easy to verify the following fact:   2 2 p11 p12   = π1 π2 + π1 (1 − π2 ) a2 N2 + π2 (1 − π1 ) a1 N1 > 0, (5.47)  |P | =  p21 p22  (1 − π1 )a1 N1 + (1 − π2 )a2 N2

provided that πi ∈ (0, 1), ai > 0, and Ni > 0, i = 1, 2.

5.3 Heterogeneous Mixing

207

Theorem 5.3 (Optimal Solution to Problem (I) When n = 2) Consider Rv = Rv (φ1 , φ2 ) as a function of φ1 and φ2 , and let η0 and κi be given in (5.46). Assume that condition (5.47) holds. (a) For a given value of η, the optimal point Φ ∗ (η) exists and lies in the interior of the unit square if and only if η0 < η < N

and

κi > 0 for i = 1, 2.

(5.48)

(b) For each η0 < η < N, the explicit formulae for P ∗ (η) and Rv{min} (η) are given by N −η (κ1 , κ2 ) κ1 N1 + κ2 N2  N −η Rv{min} (η) = |P |R01 R02 N1 N2 . κ1 N1 + κ2 N2

Φ ∗ (η) = (1, 1) −

(5.49) (5.50)

(c) If 0 < η < η0 , the minimum point Φ ∗ (η) is one of the boundary points (η/N1 , 0) or (0, η/N2 ) and hence Rv{min} (η) = min{Rv (η/N1 , 0), Rv (0, η/N2 )}. For the general case of n > 2, Theorem 5.2 can be used to derive the lower and upper bounds for the minimum reproduction number Rv{min} (η). To facilitate biological interpretations, introduce the following notation: fi := Ni /N, 1 ≤ i ≤ n Population fraction of sub-population i; n (1 − φi )fi Population fraction unvaccinated; U := ,0 := R

R0⋄ :=

i=1 n

R0i fi

i=1 n  i=1

1 fi R0i

,0 0 := minR 2 /R R 0i i

Population weighted reproduction number; (5.51) −1

Harmonic mean of R0i weighted by sub-population fractions fi ; Analogous to a scaled reproduction number.

The following results provide the lower and upper bounds for the minimum Rv{min} (η) in Problem (I): Theorem 5.4 Assume that the conditions of Theorem 5.2 hold. Let η < N, and let ,0 , R ⋄ , and R 0 be defined in (5.51). U,R 0

208

5 Models with Heterogeneous Mixing (n)

(a) The bounds of Rv{min} (η) for φ ∈ Ωp (η) ∩ [0, 1]n are 0 U ≤ Rv{min} (η) ≤ R ⋄ U . R 0

(5.52)

(b) If R0i > 1 for all i, then

η∗ 1 ≤ 1 − ⋄. N R0

(5.53)

Remarks The bounds for the optimal solutions have clear biological meanings based on the biological interpretations of the quantities in (5.51). (i) Note that R0⋄ 0 are weighted basic reproduction numbers, and the factor U is the fraction of and R the overall population that remains susceptible. In light of this, we see that the lower and upper bounds for Rv{min} (η) in (5.52) take the familiar form of an effective reproduction number. (ii) For the upper bound of η∗ , if ai = a are all the same, we have R0⋄ = R0 , in which case the upper bound in (5.53) becomes 1 − 1/R0 . This is similar to the usual formula for the critical vaccination fraction φc = 1 − 1/R0 , for which the number of vaccinated is ηc = φc N = N (1 − 1/R0 ). Although various observations about the effect of mixing on reproduction numbers have been made in previous studies, the result stated in Theorem 5.2 provided a definitive lower and upper bounds corresponding to the proportionate and the isolated mixing (a rigorous proof can be found in [43]) for a large class of mixing matrix P (not just Jacquez-type). Using a model metapopulation composed of a city and several villages, May and Anderson [36, 37] showed that heterogeneity in relevant sub-population characteristics also increased Rv . Hethcote and van Ark [28] argued that person-to-person contact rates in densely populated urban areas should be no more than twice those in sparsely populated rural ones. This change in parameter values diminished the apparent effect of heterogeneity. The facts that population heterogeneities tend to increase R0 and that models assuming proportionate mixing generate lower values of R0 have been suggested by other researchers [1, 3, 20]. Example 4 Figure 5.5 illustrates an example from [21], which extends May and Anderson’s [36] conclusion that “under a uniformly applied immunization program, the overall fraction that must be immunized is larger than would be estimated by (incorrectly) assuming the population to be homogeneously mixed.” Consider Rv = Rv (φ1 , φ2 ) as a function of vaccination coverage (φ1 , φ2 ). The two contour plots of Rv are for the cases of (a) homogeneous contacts (a1 = a2 = 10) and (b) heterogeneous contacts (a1 = 8, a2 = 12), while other parameters are the same for the two sub-populations (N1 = N2 , π1 = π2 = 0.6, σ = 0.05, γ = 1/7, θ = 1/(365 × 70)). Note that the total number of vaccine doses is given by φ1 N1 + φ2 N2 . Because N1 = N2 , a vaccination pair (φ1 , φ2 ) that minimizes the total doses if and only if it minimizes the quantity φ1 + φ2 . The thicker curve is the contour for Rv = 1. The thick dashed line with slope −1 is φ1 + φ2 = c for a

5.4 A Heterogeneous Mixing Age of Infection Model

209

Fig. 5.5 Contour plots of Rv as a function of φ1 and φ2 for (a) homogeneous population (a1 = a2 ) and (b) heterogeneous population (a1 = a2 ). For both plots, π1 = π2 and N1 = N2 . The thick solid curve represents the contour Rv = 1. The thin and thick dashed lines correspond to φ1 = φ2 and φ + φ2 = 2 × 0.74, respectively. All points (φ1 , φ2 ) on the line φ1 + φ2 = c for a constant c > 0 correspond to the same total vaccination doses. Source: [21]

constant c > 0. It shows in (a) that Rv (0.74, 0.74) = 1 and that Rv (φ1 , φ2 ) > 1 for all other pairs (φ1 , φ2 ) with φ1 + φ2 = 2 × 0.74. This suggests that the optimal allocation is the homogeneous coverage φ1 = φ2 . However, the plot in (b) shows a very different result. Particularly, among all pairs (φ1 , φ2 ) on the line φ1 + φ2 = 2 × 0.74, some can make Rv (φ1 , φ2 ) < 1. In fact, there is one point (φ1c , φ2c ) at which the minimum Rv (φ1c , φ2c ) = 0.86 is achieved. This suggests that, in a non-homogeneous population, uniform coverage (equal φi for all i) may not be the most efficient.

5.4 A Heterogeneous Mixing Age of Infection Model The basic age of infection model extends the simple SI R epidemic model by allowing an arbitrary number of stages in the model and arbitrary distributions of stay in each stage. However, it does not include the possibility of subgroups with different activity levels and heterogeneous mixing between subgroups. This possibility can be included in a heterogeneous mixing age of infection model. As in homogeneous mixing models, the age of infection approach is more general than simpler models in several respects. Age of infection models allow arbitrary distributions of stay in compartments and arbitrary sequences of compartments. In addition, they allow variable infectivity. This can be included in the kernel A(s) which leads to the infectivity function ϕ(t) describing infectivity rather than simply counting the number of infectives.

210

5 Models with Heterogeneous Mixing

As in the previous section, we consider two sub-populations of sizes N1 , N2 , respectively, each divided into susceptibles and infected members with subscripts to identify the sub-population. Suppose that Ai (s) is the mean infectivity of individuals who have been infected s time units previously, and that a1 , a2 are the contact rates of the two sub-populations. It is necessary to describe also the mixing between the two groups. Suppose that the fraction of contacts made by a member of group i that is with a member of group j is pij , i, j = 1, 2. Then p11 + p12 = p21 + p22 = 1. A two-group model may describe a population with groups differing by activity levels and possibly by vulnerability to infection, so that a1 = a2 but A1 (s) = A2 (s). It may also describe a population with one group which has been vaccinated against infection, so that the two groups have the same activity level but different disease model parameters. In this case, a1 = a2 but A1 (τ ) = A2 (τ ). In this model, any differences between groups in susceptibility or infectivity are included in the factors A1 (s), A2 (s). An age of infection model with two subgroups is Si′ = −ai Si ϕi (t) =



∞

0



pij pii ϕi + ϕj Ni Nj



[−Si′ (t − τ )]Ai (τ )dτ, i, j = 1, 2, i = j.

Here, ϕi (t) is the total infectivity of infected members of group i (i = 1, 2). As for the homogeneous mixing model, we may write this model using only the equations for Si , Si′ (t)



pii = −ai Si (t) Ni



0

∞

Ai (s)Si′ (t

pij − s)ds + Nj



∞

0

Aj (s)Sj′ (t



 ∞ N1  ∞ a1 p11 0 A1 (τ )ds a1 p12 N A (s)ds 2 0 2 P = . ∞ N2  ∞ A1 (s)ds a2 p22 0 A2 (s)ds a2 p21 N 1 0

The matrix P is similar to the matrix Q = R −1 P R, with R=

  N1 0 0 N2

− s)ds ,

i, j = 1, 2, i = j.

The next generation matrix is



(5.54)

5.4 A Heterogeneous Mixing Age of Infection Model

211

and ∞ ∞   a1 p11 0 A1 (s)ds a1 p12 0 A2 (s)ds Q= . ∞ ∞ a2 p21 0 A1 (s)ds a2 p22 0 A2 (s)ds Thus R0 is the largest root of ∞ ∞   A (s)ds a1 p11 0  A1 (s)ds − λ a1 p12 2 0 ∞ = 0. det ∞ a2 p22 0 A2 (s)ds − λ a2 p21 0 A1 (s)ds

(5.55)

In order to obtain an invasion criterion, initially when S1 (t) is close to S1 (0) = N1 and S2 (t) is close to S2 (0) = N2 , we replace S1 (t) and S2 (t) by N1 , N2 , respectively, to give a linear system, and the condition that this linear system has a solution Si (t) = Ni ert (i = 1, 2) is 1 = ai pi1



∞

e

−rs

0

A1 (s)ds + ai pi2



∞

0

e−rs A2 (s)ds, i = 1, 2.

(5.56)

The initial exponential growth rate is the solution r of the equation det



∞  ∞ −rs  A2 (s)ds a1 p11 0 e−rs A1 (s)ds − 1 a1 p12 0 e  = 0. ∞ ∞ a2 p21 0 e−rs A1 (s)ds a2 p22 0 e−rs A2 (s)ds − 1

(5.57)

In the special case of proportionate mixing, in which p11 = p21 , p12 = p22 , so that p12 p21 = p11 p22 , the basic reproduction number is given by R0 = a1 p11



0

∞

A1 (s)ds + a2 p22



∞

A2 (s)ds,

0

and Eq. (5.57) reduces to a1 p11



∞

e

0

−rs

A1 (s)ds + a2 p22



∞ 0

e−rs Ai (s)ds = 1.

(5.58)

There is an epidemic if and only if R0 > 1. In the special case in which the two groups have the same infectivity distribution but may have different activity levels and possibly vulnerability to infection, so that A1 (s) = A2 (s) = A(s), R0 is the largest root of det



∞ ∞  a1 p11 0  A(s)ds − λ a1 p12 0 A(s)ds  ∞ ∞ a2 p21 0 A(s)ds a2 p22 0 A(s)ds − λ

212

5 Models with Heterogeneous Mixing

and the initial exponential growth rate is the solution r of the equation det



∞  ∞ −rs  A(s)ds a1 p11 0 e−rs A(s)ds − 1 a1 p12 0 e  = 0. ∞ ∞ a2 p21 0 e−rs A(s)ds a2 p22 0 e−rs A(s)ds − 1

Comparing Eqs. (5.55) and (5.59), we see that each of R0 /  ∞ −rτ 1/ 0 e A(τ )dτ is the largest eigenvalue of the matrix

∞ 0

(5.59)

A(τ )dτ and

  a1 p11 a1 p12 , a2 p21 a2 p22

the largest root of the equation x 2 − (a1 p11 + a2 p22 )x + a1 a2 (p11 p22 − p12 p21 ) = 0. Thus ∞ 0

R0 1 =  ∞ −rs , A(s)ds A(s)ds 0 e

which implies the same relation as for the homogeneous mixing model. Thus, if we assume heterogeneous mixing, we obtain the same estimate of the reproduction number from observation of the initial exponential growth rate, and this conclusion remains valid for an arbitrary number of groups with different contact rates. The estimate of the basic reproduction number from the initial exponential growth rate does not depend on heterogeneity of the model. This result does not generalize to the case A1 (s) = A2 (s), but it does remain valid for an arbitrary number of groups with different contact rates.

5.4.1 The Final Size of a Heterogeneous Mixing Epidemic With homogeneous mixing, knowledge of the basic reproduction number translates into knowledge of the final size of the epidemic. However, with heterogeneous mixing, even in the simplest case of proportionate mixing, the size of the epidemic is not determined uniquely by the basic reproduction number. For the heterogeneous mixing model (5.54) there is a pair of final size relations. We divide the equation for S1 in (5.54) by Si (t) and integrate with respect to t from 0 to ∞. Much as in the derivation of the final size relation for the homogeneous

5.4 A Heterogeneous Mixing Age of Infection Model

213

mixing model we obtain a pair of final size relations which may be solved for S1 (∞) and S2 (∞): log

  2   ∞ pij

Si (0) ai = Aj (s)ds , Nj − Sj (∞) Si (∞) Nj 0 j =1

i = 1, 2.

(5.60)

The system of equations (5.60) has a unique solution (S1 (∞), S2 (∞)). In order to prove this, we define gi (x1 , x2 ) = log

  ∞ 2 xj Si (0) − ai pij 1 − Aj (s)ds. xi Nj 0 j =1

A solution of (5.60) is a solution (x1 , x2 ) of the system gi (x1 , x2 ) = 0,

i = 1, 2.

For each x2 , g1 (0+ , x2 ) > 0, g1 (S1 (0), x2 ) < 0. Also, as a function of x1 , g1 (x1 , x2 ) either decreases or decreases initially and then increases to a negative value when x1 = S1 (0). Thus for each x2 < S2 (0), there is a unique x1 (x2 ) such that g1 (x1 (x2 ), x2 ) = 0. Also, since g1 (x1 , x2 ) is an increasing function of x2 , the function x1 (x2 ) is increasing. Now, since g2 (x1 , 0+ ) > 0, g2 (x1 , S2 (0)) < 0, there exists x2 such that g2 (x1 (x2 ), x2 ) = 0. Also, g2 (x1 (x2 ), x2 ) either decreases monotonically or decreases initially and then increases to a negative value when x2 = S2 (0). Therefore this solution is also unique. This implies that (x1 (x2 ), x2 ) is the unique solution of the final size relations. Numerical simulations indicate that models with heterogeneous mixing may give very different epidemic sizes than models with the same basic reproduction number and homogeneous mixing. The reproduction number of an epidemic model is not sufficient to determine the size of the epidemic if there is heterogeneity in the model. We conjecture that for a given value of the basic reproduction number the maximum epidemic size for any mixing is obtained with homogeneous mixing. ∞ Assume that the parameters N1 , N2 , 0 A(τ )dτ remain fixed and attempt to minimize S1 (∞) + S2 (∞) as a function of a1 , a2 (with a1 , a2 constrained to keep p1 a1 + p2 a2 = k fixed and p1 , p2 as specified by proportionate mixing). Homogeneous mixing corresponds to a1 = a2 . The constraint relating a1 , a2 implies that 2a1 − k N1 da2 = · ; da1 k − 2a2 N2

214

5 Models with Heterogeneous Mixing

when a1 = a2 = k we have da2 N1 =− . da1 N2 Also, when a1 = a2 , p1 =

N1 , N1 + N2

p2 =

N2 , N1 + N2

S1 S2 = , N1 N2

dp1 N1 . = da1 kN

If we differentiate with respect to a1 , we can calculate that d[S1 (∞) + S2 (∞)] =0 da1 when a1 = a2 . We believe that a1 = a2 is the only critical point of S1 (∞) + S2 (∞), although we have not been able to verify this analytically. If a1 = a2 is the only critical point of S1 (∞) + S2 (∞), this critical point must be a minimum. We conjecture that this result is also valid if we allow arbitrary mixing, that is, we conjecture that for a given value of the basic reproduction number the maximum epidemic size for any mixing is obtained with homogeneous mixing. While we have confined the description of the heterogeneous mixing situation to a two-group model, the extension to an arbitrary number of groups is straightforward. We suggest that in advance planning for a pandemic, the number of groups to be considered for different treatment rates should determine the number of groups to be used in the model. On the other hand, the number of groups to be considered should also depend on the amount and reliability of data, and these two criteria may be contradictory. A model with fewer groups and parameters chosen as weighted averages of the parameters for a model with more groups may give predictions that are quite similar to those of the more detailed models. We suggest also that use of the final size relations for a model with total population size assumed constant is a good time-saving procedure for making predictions if the disease death rate is small. We have seen that in the case of homogeneous mixing, knowledge of the initial exponential growth rate and the infective period distribution is sufficient to determine the basic reproduction number and thence the final size of an epidemic. In the case of heterogeneous mixing, knowledge of the initial exponential growth rate and the infective period distribution is sufficient to determine the basic reproduction number, but not to determine the final size of the epidemic. This raises the question of what additional information that may be measured at the start of a disease outbreak would suffice to determine the epidemic final size if the mixing is heterogeneous. We assume that A1 (s), A2 (s), and the mixing matrix   p11 p12 M= p21 p22

5.4 A Heterogeneous Mixing Age of Infection Model

215

are known. The next generation matrix is 

∞ a1 pN111 0 A1 (s)ds ∞ K= a2 pN211 0 A1 (s)ds

 ∞ a1 pN122 0 A2 (s)ds ∞ , a2 pN222 0 A2 (s)ds

and R0 is the largest (positive) eigenvalue of this matrix. There is a corresponding eigenvector with positive components u=

  u1 . u2

Since the components of this eigenvector give the proportions of infective cases in the two groups initially, it is reasonable to hope to be able to determine this eigenvector from early outbreak data. The general final size relation is log

  2   Sj (∞)  ∞ Si (0) ai pij 1 − = Aj (s)ds , Si (∞) Nj 0 j =1

i = 1, 2.

These equations may be solved for S1 (∞), S2 (∞) if the contact rates a1 , a2 can be determined from the available information. The condition that the vector u with components (u1 , u2 ) is an eigenvector of the next generation matrix corresponding to the eigenvalue R0 is

 ai pi1 u1 + pi2 u2



∞ 0

Ai (s)ds = R0 ui ,

i = 1, 2,

and since it is assumed that the function A(τ ), the vector u, and the mixing matrix (pij ) are known these equations determine a1 and a2 . In vector notation, if we define the column vector   a a= 1 a2 and the row vectors

we have

  M j = pj 1 p j 2 , aj =

R0 ∞ uj . Mj u 0 Aj (s)ds

216

5 Models with Heterogeneous Mixing

When these values are substituted into the final size system, S1 (∞) and S2 (∞) may be determined. This argument extends easily to models with an arbitrary number of activity groups. In real-life applications, there are usually many groups, and the final size of an epidemic is obtained most efficiently by numerical simulations. The results obtained here are more likely to be useful in theoretical applications, such as comparisons of different control strategies. One may think of the case a1 = a2 , A1 (s) = A2 (s) as a model for a disease with heterogeneous mixing but not treatment and the case a1 = a2 , A1 (s) = A2 (s) as a model for a disease in which the mixing is homogeneous but treatment that changes the infective period distribution has been applied to a part of the population. Of course, if the treatment also includes quarantine that also changes the contact rate, the case a1 = a2 , A1 (s) = A2 (s) would be appropriate. We suggest that in advance planning for a pandemic, the number of groups to be considered for different treatment rates should determine the number of groups to be used in the model. On the other hand, the number of groups to be considered should also depend on the amount and reliability of data, and these two criteria may be contradictory. A model with fewer groups and parameters chosen as weighted averages of the parameters for a model with more groups may give predictions that are quite similar to those of the more detailed models. We suggest also that use of the final size relations for a model with total population size assumed constant is a good time-saving procedure for making predictions if the disease death rate is small.

5.5 Some Warnings An actual epidemic differs considerably from the idealized models such as (5.1) as well as the extensions considered later. Some notable differences are: 1. When it is realized that an epidemic has begun, individuals are likely to modify their behavior by avoiding crowds to reduce their contacts and by being more careful about hygiene to reduce the risk that a contact will produce infection. 2. If a vaccine is available for the disease which has broken out, public health measures will include vaccination of part of the population. Various vaccination strategies are possible, including vaccination of health care workers and other first line responders to the epidemic, vaccination of members of the population who have been in contact with diagnosed infectives, or vaccination of members of the population who live in close proximity to diagnosed infectives. 3. Diagnosed infectives may be hospitalized, both for treatment and to isolate them from the rest of the population. Isolation may be imperfect; in-hospital transmission of infection was a major problem in the SARS epidemic. 4. Contact tracing of diagnosed infectives may identify people at risk of becoming infective, who may be quarantined (instructed to remain at home and avoid

5.6 *Projects: Reproduction Numbers for Discrete Models

217

contacts) and monitored so that they may be isolated immediately if and when they become infective. 5. In some diseases, exposed members who have not yet developed symptoms may already be infective, and this would require inclusion in the model of new infections caused by contacts between susceptibles and asymptomatic infectives from the exposed class. 6. In the SARS epidemic of 2002–2003 in-hospital transmission of disease from patients to health care workers or visitors because of imperfect isolation accounted for many of the cases. This points to an essential heterogeneity in disease transmission which must be included whenever there is any risk of such transmission.

5.6 *Projects: Reproduction Numbers for Discrete Models This project concerns the computation of the reproduction number for discrete models using the approach of the next generation matrix. A formula for the reproduction number R (either R0 or RC ) is derived by adopting the method used in [2] based on the next generation matrix approach. That is, in the discrete-time case R = ̺(F (I − T )−1 ),

(5.61)

where ̺ represents the spectral radius, F is the matrix associated with new infections, and T is the matrix of transitions with ̺(T ) < 1 (see [2, 16, 31, 48]). Here F and T are calculated on the infected variables only evaluated at the diseasefree equilibrium, and the Jacobian on these variables is F + T , which is assumed to be irreducible. Consider the simple discrete SEIR model with geometric distributions for the latent and infectious period with parameters α and δ (α < 1, δ < 1), respectively. This is equivalent to assuming constant transition probabilities 1 − α and 1 − δ per unit time from E to I and from I to R, respectively. The model reads In

Sn+1 = Sn e−β N ,

In

En+1 = Sn (1 − e−β N ) + αEn In+1 = (1 − α)En + δIn ,

(5.62)

n = 1, 2, · · · .

For system (5.62), the matrices associated with new infections and transitions are   0β F = 00

and



 α 0 T = , 1−α δ

218

5 Models with Heterogeneous Mixing

respectively. Then ̺(T ) = max{α, δ} < 1; thus, R = ̺(F (I − T )−1 ) =

β . 1−δ

(5.63)

Question 1 Extend the model (5.62) by incorporating isolation or hospitalization of infectious individuals. For the extended model, compute the reproduction number using the formula (5.61). Question 2 Extend the model (5.62) by considering different transmission rates βi for individuals in different infectious stages Ii (i = 1, 2, · · · ). Use the formula (5.61) to compute the reproduction number for the extended model. Project 2 Consider next the case when the infective period follows an arbitrary discrete (bounded) distribution, which is denoted by Y . Let qi = P(Y > i) and P(Y = i) = qi−1 − qi . It is easy to see that qi is a decreasing function, i.e., qi ≥ qi+1 . In fact, q0 = 1 and qm = 0 for all m ≥ M, where M is the maximum number of units of time that an individual takes to recover. Because the geometric is the only memoryless discrete distribution, when other distributions are considered it is necessary to keep track of the past in order to know the values at the present. In fact, it is impossible to use the next generation matrix approach directly because the disease stages (S, E, and I ) at time n + 1 cannot be written in the form   [ En+1 , In+1 , Sn+1 ]T = M [ En , In , Sn ]T , where M : R3 → R3 . To overcome this difficulty we can consider multiple I stages, similar to the approach known as the “linear chain trick” used in continuous models to convert a gamma distribution to a sequence of exponential distributions. Thus, we introduce the subclasses I (1) , I (2) , · · · , I (M) (see Fig. 5.6). The superscript i corresponds to the time since becoming infectious. Notice that these subclasses I (i) are different from those in the negative binomial model because

Fig. 5.6 A transition diagram for the case when the stage duration of the infective period has an arbitrary bounded distribution with upper bound M. The superscript i is the stage age in the infectious period and individuals in I (i) (for all i) can enter the recovered class R with a certain probability

5.6 *Projects: Reproduction Numbers for Discrete Models

219

here an individual can only stay in I (i) for one unit of time, and must go to either the I (i+1) class with probability qi or the recovered class R with probability 1 − qi . From Fig. 5.6 the model equations can be written as

M

(i) In

Sn+1 = Sn e− i=1 βi N ,  (i) 

M In En+1 = Sn 1 − e− i=1 βi N + αEn , (1)

(2)

(1)

(5.64)

(j )

In+1 = (1 − α)En , In+1 = q1 In , In+1 =

qj −1 (j −1) , qj −2 In

3 ≤ j ≤ M,

where βi denote the transmission rates at the infective stage i, 1 ≤ i ≤ M. As qi is the probability that an infective individual remains infective i time units after (2) (3) becoming infective, the transition probability from In to In+1 is given by the probability that an infective individual is still infective two time units after becoming infectious given that the person remained infective one time unit ago, i.e., q2 /q1 . (j ) (3) equation and similarly In+1 equations for 3 ≤ j ≤ M. This explains the In+1 Question 1 For the case when transmission rates βi are stage-dependent, show that −1

R0 = ̺(F (I − T )

)=

M

(5.65)

βi qi−1 .

i=1

Question 2 Derive R0 from its biological definition. Hint: Using the fact that the distribution Y has an upper bound M and that for a given function f M

m=1

P(Y = m)f (m) = E(f (Y )).

The reproduction number from the biological definition (with f (m) = M−1

R= βi+1 qi .

m

i=1 βi )

is

i=0

The formula (5.63) can also be applied to models with various heterogeneities. Consider a model that includes two sub-populations, female and male populations, with heterogeneous mixing (i.e., no sexual contacts between individuals of the same sex). Assume that the infective periods for female and male populations follow arbitrary discrete (bounded) distributions denoted by Yf and Ym , respectively. Here the subscripts f and m stand for female and male, respectively. Let qf,i = P(Yf > i) and qm,i = P(Ym > i) with qf,0 = qm,0 = 1, and the upper bounds for the two distributions (i.e., the maximum numbers of units of time that an individual takes to recover) be Mw for w = f, m.

220

5 Models with Heterogeneous Mixing

The model equations are Sw,n+1 = Sw,n e−

Mw˜

i=1

(i)

βw,i ˜ Iw,n ˜ /N

,

Mw˜ (i)   ˜ Iw,n ˜ /N + α E Ew,n+1 = Sw,n 1 − e− i=1 βw,i w w,n , (1) (2) (1) Iw,n+1 = (1 − αw )Ew,n , Iw,n+1 = qw,1 Iw,n , (j )

Iw,n+1 =

qw,j −1 (j −1) Iw,n , qw,j −2

3 ≤ j ≤ Mw ,

for w = f, m.

Here w˜ represents the opposite sex of w, i.e., f˜ = m, m ˜ = f . The constant βf˜,i (βm,i ) represents the infection rate to a female (male) transmitted by infectious male ˜ (female) individuals with stage age i. Question 3 Show that the reproduction number is given by . Mf   . Mm . −1 / R = ̺(F (I − T ) ) = βf,i qf,i−1 βm,i qm,i−1 . i=1

(5.66)

i=1

The square root in (5.66) is a consequence of the fact that the secondary infections need to be computed from one female (male) to other females (males) through the male (female) population. Hint: Consider the order of variables: (1)

(M )

(2)

(1) (2) (Mm ) (Ef,n , If,n , If,n , · · · , If,nf , Em,n , Im,n , Im,n , · · · , Im,n ).

First show that F (I − T )−1 =



 0 Fm (I − Tm )−1 , Ff (I − Tf )−1 0

where

Fw (I − Tw )−1

⎡ Mw Mw ⎢ β q βw,i qw,i−1 w,i w,i−1 ⎢ ⎢ i=1 i=1 ⎢ =⎢ 0 0 ⎢ .. .. ⎢ ⎣ . . 0 0

References: [2, 16, 31, 48].

⎤

· · · βw,Mw ⎥ ⎥ ⎥ ⎥ ··· 0 ⎥, ⎥ .. ⎥ .. . . ⎦ ··· 0

w = f, m.

5.7 *Project: Modeling the Synergy Between HIV and HSV-2

221

5.7 *Project: Modeling the Synergy Between HIV and HSV-2 Consider the following model for HSV-2, which includes a male population (specified by subscript m) and a female population with two sub-groups representing low-risk and high-risk groups (specified by subscripts f1 and f2 , respectively): dSi = μi Ni − λi (t)Si − μi Si , dt dAi = λi (t)Si + γi (θi )Li − (ωi + θi + μi )Ai , dt

 dLi = (ωi + θi )Ai − γi (θi ) + μi Li , i = m, f1 , f2 , dt

(5.67)

where λi (t) (i = m, f1 , f2 ) are the force of infection functions given by λm (t) =

2

bm ci βfi m

i=1

Am , λfj (t) = bfj βmfj Nm

Afi , Nfi

(5.68)

j = 1, 2,

and Ni = Si + Ai + Li , i = m, f1 , f2 . Each group i (i = m, f1 , f2 ) is divided into three subgroups: susceptible (Si ), infected with acute HSV-2 only (Ai ), infected with latent HSV-2 only (Li ). The population within each group is assumed to be homogeneous in the sense that individuals have the same infective period, duration of immunity, contact rate, and so on. A transition diagram between these epidemiological classes within group i is depicted in Fig. 5.7. For each sub-population i (i = f1 , f2 , m) there is a per-capita recruitment rate μi into the susceptible group. For all classes there is a constant per-capita rate μi of exiting the sexually active population. Thus, the total population Ni in group i remains constant for all time. Susceptible people in group i acquire infection with HSV-2 at the rate λi (t). Upon being infected with HSV-2, people in group i enter the class Ai . These individuals become latent Li at the constant rate ωi (an average duration in Ai is 1/ωi ). Following an appropriate stimulus in individuals with latent HSV-2, reactivation may occur at the rate γi . Finally, the antiviral treatment rate for the Ai individuals is denoted by θi . Because antiviral medications will also suppress

Fig. 5.7 A transition diagram for HSV-2 for subgroup i (i = m, f1 , f2 )

222

5 Models with Heterogeneous Mixing

reactivation of latent HSV-2, we assume that the reactivation rate of people with latent HSV-2 γi is a decreasing function of θi , denoted by γi (θi ). For the forces of infection λi (t) (i = m, f1 , f2 ), bi is the rate at which individuals in group i acquire new sexual partners (also referred to as contact rates), and cj denotes the probability that a male chooses a female partner in group j (j = f1 , f2 ). Then c1 + c2 = 1. For ease of notation, let c1 = c,

c2 = 1 − c.

Overall, the number of female partners in groups j (j = 1, 2) that males acquire should be equal to the number of male partners that females in groups j acquire. These observations lead to the following balance conditions: bm (1 − c)Nm = bf2 Nf2 .

bm cNm = bf1 Nf1 ,

(5.69)

To ensure that constraints in (5.69) are satisfied, we assume in numerical simulations that bm and c are fixed constants with bf1 and bf2 being varied according to Nm , Nf1 , and Nf2 . The parameters βim (βmi ), i = f1 , f2 are the HSV-2 transmission probabilities per partner between females infected with acute HSV-2 in group i and susceptible males (between males infected with acute HSV-2 and susceptible females in group i). Question 1 Let Rmfj m denote the average number of secondary male infections generated by one male individual through females in group fj (j = 1, 2). Show that " bfj βmfj bm cj βfj m Rmfj m = · Pm · · Pfj , j = 1, 2 ωm + θm + μm ωfj + θfj + μfj with Pi (i = m, f1 , f2 ) representing the probability that an individual of group i is in the acute stage (A), which is given by 



ωi + θi + μi γi (θi ) + μi  , Pi =  L γi (θi ) + ωi + θi + μi μi

i = m, f1 , f2 .

(5.70)

Question 2 Let R denote the overall reproduction number for the entire population. (a) Show that R=

0

 2 2  Rmf1 m + Rmf2 m ,

where Rmfj m (j = 1, 2) are given in Question 1.

(5.71)

5.8 Project: Effect of Heterogeneities on Reproduction Numbers

223

(b) Provide a biological interpretation of the expression on the right-hand side of Eq. (5.71). Question 3 Let E0 denote the disease-free equilibrium of the system (5.67), and let E ∗ denote an endemic equilibrium. (a) Show that E0 is locally asymptotically stable when R < 1 and unstable when R > 1. (b) Choose the function for γ (θ ) to be in the following form: γi (θi ) = γi (0)αi /(αi + θi ). Show via numerical simulations that E ∗ exists and is locally asymptotically stable when R > 1. One case to consider is when c > 0.5, e.g., c = 0.9 (90% of male contacts are with the low-risk female group). Because of the constraint (5.69), bi and Ni are not independent. Choose bm = 0.1, bf1 = 0.0901, bf2 = 9.01 (so that bf2 /bf1 = 100), Nm = Nf1 + Nf2 = 107 (e.g., Nf1 = 9.9889 × 106 , Nf2 = 1.1099 × 104 ). Consider the case when treatment is absent, i.e., θ = 0. Other parameter values are ω = 2.5, γm (0) = 0.436, γf1 = γf2 = 0.339, αi = 2. The time unit is month. (c) Explore numerically the effect of treatment θ . Consider various scenarios such as treatment in only one subgroup (male, low-risk female or high-risk female group). Summarize the observed outcomes in terms of effect of treatment on the prevalence of HSV-2.

5.8 Project: Effect of Heterogeneities on Reproduction Numbers Consider the metapopulation model (5.37), which includes vaccination coverage φ = (φ1 , φ2 , · · · , φn ) in the n sub-populations. As pointed out in Sect. 5.3.1 that several types of heterogeneities including the activity (ai ), sub-population size (Ni ), and preference for mixing within the sub-population (πi ) may affect the optimal vaccination strategy. In this project, we examine in more details how these heterogeneities may affect Rv for the case of n = 2, and how to choose (φ1 , φ2 ) to reduce Rv below a certain level. For example, Fig. 5.8 illustrates the different parameter regions in the (φ1 , φ2 ) plane in which Rv < 1 in the cases of (a) proportionate mixing (π1 = π2 = 0) and (b) preferential mixing (πi > 0). For Questions 1–3 below, let σ = 0.05, γ = 1/7. Question 1 For the cases in (a) and (b) below, determine the values of the basic reproduction number R0 for each case. Describe how preferential mixing and heterogeneous activity may influence the effect of heterogeneity in activity on R0 . Let N1 = N2 = 500. Determine the values of R0 for each case.

224

5 Models with Heterogeneous Mixing

Fig. 5.8 Plots of Rv as a function of φ1 and φ2 for (a) proportionate mixing (πi = 0, no preference) and (b) preferential mixing (πi > 0). Values at or below their intersection with the dark plane, Rv = 1, are combinations of φi (i = 1, 2) at which population-immunity attains or exceeds this threshold

(a) Homogeneous activity: a1 = a2 = 10 (a1 + a2 = 20). (i) No preference: π1 = π2 = 0. (ii) Homogeneous preference: π1 = π2 = 0.5. (iii) Heterogeneous preference: π1 = 0.25, π2 = 0.75 and π1 = 0.75, π2 = 0.25. (b) Heterogeneous activity: a1 = 8 and a2 = 12 (a1 + a2 = 20). Repeat (i)–(iii) in (a). Question 2 Same as in Question 1 but consider heterogeneities in both activity ai and population size Ni . Given that N1 + N2 = N = 1000 and a1 = 5, a2 = 10. Repeat (i)–(iii) in Question 1(a) for the following two cases: (a) Homogeneous population size: N1 = N2 = 0.5N. (b) Heterogeneous population size: N1 = 0.1N and N2 = 0.9N. Question 3 Consider the effective reproduction number Rv (φ1 , φ2 ) as a function of vaccination coverage (φ1 , φ2 ). (a) Find the optimal solution Φ ∗ = (φ1∗ , φ2∗ ) for the following set of parameter values: π1 = π2 = 0.3, a1 = 15, a2 = 12, N1 = 1100, N2 = 900. The given number of vaccine doses is φ1 N1 + φ2 N2 = 990. (b) What is the minimum value Rv{min} = Rv (Φ ∗ )?

References

225

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24. Gani, R., H. Hughes, T. Griffin, J. Medlock, & S. Leach (2005) Potential impact of antiviral use on hospitalizations during influenza pandemic, Emerg. Infect. Dis. 11: 1355–1362. 25. Glasser, J., Z. Feng, A. Moylan, S. Del Valle & C. Castillo-Chavez (2012) Mixing in agestructured population models of infectious diseases, Math. Biosci., 235(1): 1–7. 26. Glasser, J.W., Z. Feng, S.B. Omer, P.J. Smith, L.E. Rodewald (2016) The effect of heterogeneity in uptake of the measles, mumps, and rubella vaccine on the potential for outbreaks of measles: a modelling study, 16(5): 599–605. 27. Hethcote, H.W. & J.A. Yorke (1984) Gonorrhea Transmission Dynamics and Control, Lect. Notes in Biomath. 56, Springer-Verlag, Berlin-Heidelberg-New York (1984). 28. Hethcote, H.W. & J.W. van Ark (1987) Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation and immunization programs. Math. Biosci., 84(1): 85–118. 29. Hirsch, M.W. & S. Smale, Differential Equations (1974) Dynamical Systems, and Linear Algebra, Academic Press, Orlando, FL (1974). 30. Jacquez, J.A., C.P. Simon, J. Koopman, L. Sattenspiel, & T. Perry (1988) Modeling and analyzing HIV transmission: the effect of contact patterns, Math. Biosci., 92: 119–199. 31. Lewis, M.A., J. Renclawowicz, P. van Den Driessche, and M. Wonham (2006) A comparison of continuous and discrete-time West Nile Virus models, Bull. Math. Biol. 68(3): 491–509. 32. Longini, I.M., M.E. Halloran, A. Nizam, & Y. Yang (2004) Containing pandemic influenza with antiviral agents, Am. J. Epidem. 159: 623–633. 33. Longini, I.M., A. Nizam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D.A.T. Cummings, & M.E. Halloran (2005) Containing pandemic influenza at the source, Science 309, 1083–1087. 34. Longini, I. M. & M. E. Halloran (2005) Strategy for distribution of influenza vaccine to high risk groups and children, Am. J. Epidem. 161: 303–306. 35. Ma, J. & D.J.D. Earn (2006) Generality of the final size formula for an epidemic of a newly invading infectious disease, Bull. Math. Biol. 68: 679–702. 36. May, R.M. & R.M. Anderson (1984) Spatial heterogeneity and the design of immunization programs. Math Biosci. 72(1): 83–111. 37. May, R.M. & R.M. Anderson (1984) Spatial, temporal and genetic heterogeneity in host populations and the design of immunization programmes. IMA J Math Appl Math Biol. 1(3): 233–66. 38. Meyers, L.A. (2007) Contact network epidemiology: Bond percolation applied to infectious disease prediction and control, Bull. Am. Math. Soc. 44, 63–86. 39. Meyers, L.A. , M.E.J. Newman & B. Pourbohloul (2006) Predicting epidemics on directed contact networks, J. Theor. Biol. 240, 400–418. 40. Meyers, L.A., B. Pourbohloul, M.E.J. Newman, D.M. Skowronski, & R.C. Brunham (2005) Network theory and SARS: predicting outbreak diversity. J. Theor. Biol., 232: 71–81. 41. Mossong, J., N. Hens, M. Jit, P. Beutels, K. Auranen, R. Mikolajczyk, . . . & W. J. Edmunds (2008) Social contacts and mixing patterns relevant to the spread of infectious diseases, PLoS Medicine, 5(3): e74. 42. Nold, A. (1980) Heterogeneity in disease transmission modeling, Math. Biosci, 52: 227–240. 43. Poghotanyan, G., Z. Feng, J.W. Glasser, A.N. Hill (2018) Constrained minimization problems for the reproduction number in meta-population models, J. Math. Biol. https:/doi.org/10.1007/ s00285-018-1216-z. 44. Pourbohloul, B. & J. Miller (2008) Network theory and the spread of communicable diseases, Center for Disease Modeling Preprint 2008-03, www.cdm.yorku.ca/cdmprint03.pdf 45. van den Driessche, P. & J. Watmough (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosc. 180: 29–48. 46. van den Driessche, P. & J. Watmough (2002) Further notes on the basic reproduction number, in Mathematical Epidemiology, F. Brauer, P. van den Driessche, & J. Wu (eds.) Springer Lecture Notes, Vol. 1945.

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Chapter 6

Models for Diseases Transmitted by Vectors

6.1 Introduction Many diseases are transmitted from human to human indirectly, through a vector. Vectors are living organisms that can transmit infectious diseases between humans. Many vectors are bloodsucking insects that ingest disease-producing microorganisms during blood meals from an infected (human) host, and then inject it into a new host during a subsequent blood meal. The best known vectors are mosquitoes for diseases including malaria, dengue fever, chikungunya, Zika virus, Rift Valley fever, yellow fever, Japanese encephalitis, lymphatic filariasis, and West Nile fever, but ticks (for Lyme disease and tularemia), bugs (for Chagas’ disease), flies (for onchocerciasis), sandflies (for leishmaniasis), fleas (for plague, transmitted by fleas from rats to humans), and some freshwater snails (for schistosomiasis) are vectors for some diseases. Every year there are more than a billion cases of vector-borne diseases and more than a million deaths. Vector-borne diseases account for over 17% of all infectious diseases worldwide. Malaria is the most deadly vector-borne diseases and caused an estimated 627,000 deaths in 2012. The most rapidly growing vector-borne disease is dengue, for which the number of cases has multiplied by 30 in the last 50 years. These diseases are found more commonly in tropical and sub-tropical regions where mosquitoes flourish and in places where access to safe drinking water and sanitation systems is uncertain. Some vector-borne diseases, such as dengue, chikungunya, and West Nile virus , are emerging in countries where they were unknown previously because of globalization of travel and trade and environmental challenges such as climate change. Many of the important underlying ideas of mathematical epidemiology arose in the study of malaria begun by Sir. R.A. Ross [16]. Malaria is one example of a disease with vector transmission, the infection being transmitted back and forth between vectors (mosquitoes) and hosts (humans). It kills hundreds of thousands of © Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_6

229

230

6 Models for Diseases Transmitted by Vectors

people annually, mostly children and mostly in poor countries in Africa. Among communicable diseases, only tuberculosis causes more deaths. We will analyze some models for malaria in a later chapter. Other vector diseases include West Nile virus and HIV with heterosexual transmission, yellow fever, and dengue fever (which will also be studied in a later chapter). Vector-transmitted diseases require models that include both vectors and hosts. For most diseases transmitted by vectors, the vectors are insects, with a much shorter life span than the hosts, who may be humans as for malaria or animals as for West Nile virus. For heterosexually transmitted human diseases, transmission goes back and forth between males and females rather than between two different species, but a model still requires two separate groups. The compartmental structure of the disease may be different in host and vector species; for many diseases with insects as vectors an infected vector remains infected for life so that the disease may have an SI or SEI structure in the vectors and an SI R or SEI R structure in the hosts. We will describe vector models with SEI R structure in the host species and SEI structure in the vector species, but the analysis of other types of vector-transmitted diseases is similar. The models are of the same type as those studied in Chaps. 2–4, but have a more complicated structure because they involve two different species.

6.2 A Basic Vector Transmission Models Ross received the second Nobel Prize in Medicine for demonstrating the vector transmission nature of malaria, and he then constructed a model in 1909 that predicted the possibility of controlling malaria by decreasing the mosquito population size below a threshold value. This prediction was borne out in practice, but controlling the mosquito population size is very difficult because of the ability of mosquitoes to adapt to pesticides. Malaria remains a very dangerous disease. We describe a basic model for a vector-transmitted disease, in which we assume throughout that vectors satisfy a simple SEI model, with no recovery from infection, and the hosts satisfy a simple SEI R model. This model is a template for vector disease transmission models for many specific diseases. We are thinking of mosquitoes as vectors, and because a mosquito lifetime is much shorter than that of the human hosts we must include demographics in the vector population. The model includes both epidemic and endemic situations. We consider a constant total population size Nh of hosts (humans), divided into Sh susceptibles, Eh exposed members, Ih infectives, and Rh removed members. There is a birth rate Λh in the susceptible class and a proportional natural death rate μh in each class. Exposed hosts proceed to the infective class at rate ηh and infected hosts recover at rate γ . There is a constant birth rate μNv of vectors in unit time and a proportional vector death rate μ in each class, so that the total vector population size Nv is constant. The vector population is divided into Sv susceptibles, Ev exposed members, and Iv

6.2 A Basic Vector Transmission Models

231

infectives. Exposed vectors move to the infected class at rate ηv and do not recover from infection. For simplicity, we assume that there are no disease deaths of either hosts or vectors. We assume that the mosquito biting rate is proportional to the size Nh of the host population. Thus an average mosquito makes bNh bites in unit time. The total number of mosquito bites in unit time is bNh Nv and the number of bites received by an average host in unit time is bNv . We assume that fvh is the probability that a bite transmits infection from vector to host and fhv is the probability that a bite transmits infection from host to vector. We define βh = bfvh Nv ,

βv = bfhv Nh .

If we eliminate b from these two equations, we obtain a balance relation fvh βh Nv = fhv βv Nh .

(6.1)

In many models for vector-transmitted diseases, the vector population size is much larger than the host population size but the two contact rates βh and βv are of the same order of magnitude, suggesting that fhv >> fvh . Then the number of new infective hosts in unit time is bfvh Nv Sh

Iv Iv = βh Sh . Nv Nv

A similar argument shows that the number of new mosquito infections is βv Sv

Ih . Nh

The model is Sh′ = Λ(Nh ) − βh Sh Eh′ = βh Sh

Iv − μh Sh Nv

Iv − (ηh + μh )Eh Nv

Ih′ = ηh Eh Iv − (γ + μh )Ih Sv′ = μv Nv − βv Sv Ev′ = βv Sv

Ih − μv Sv Nh

Ih − (ηv + μv )Ev Nh

Iv′ = ηv Ev − μv Iv . The corresponding epidemic model is (6.2) with Λ(Nh ) = μh = 0.

(6.2)

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6 Models for Diseases Transmitted by Vectors

6.2.1 The Basic Reproduction Number The basic reproduction number is defined as the number of secondary disease cases caused by introducing a single infective host into a wholly susceptible population of both hosts (humans) and vectors (mosquitoes). For the model (6.2) this may be calculated directly. There are two stages. First, the infective human infects mosquitoes, at a rate βv for a time 1/(γ + μh ). This produces βv /(γ + μh ) infected mosquitoes, of whom a fraction ηv /(ηv + μv ) proceeds to become infective. The second stage is that these infective mosquitoes infect humans at a rate βh for a time 1/μv , producing βh /μv infected humans per mosquito. The net result of these two stages is βh ηv βv ηv = βv βh γ + μh ηv + μv μv (ηv + μv )(γ + μh )μv infected humans, and this is the basic reproduction number R0 . We could also calculate the basic reproduction number by using the next generation matrix approach [21]. This would give the next generation matrix K



0

1 βh γ +μ h

 βv μv (μηvv+ηv ) . 0

The basic reproduction number is the positive eigenvalue of this matrix, R0 =

0

βh βv

ηv . μv (γ + μh )(μv + ηv )

In this calculation, the transition from host to vector to host is considered as two generations. In studying vector-transmitted diseases it is common to consider this as one generation and use the value that we obtained by our direct approach R0 = βh βv

ηv . μv (γ + μh )(μv + ηv )

(6.3)

This choice is made in [4, 10] and is the choice that we make because it conforms to the result obtained by direct calculation, without using the next generation approach. However, other references, including [13], use the square root form, and it is important to be aware of which form is being used in any study. The two choices have the same threshold value. In fact, different expressions are possible for the next generation matrix. This is shown in [6]. Using the next generation matrix approach but considering only host infections as new infections and vector infections as transitions, we would obtain the form (6.3).

6.2 A Basic Vector Transmission Models

233

6.2.2 The Initial Exponential Growth Rate In order to determine the initial exponential growth rate from the model, a quantity that can be compared with experimental data, we linearize the model (6.2) about the disease-free equilibrium Sh = Nh , Eh = Ih = 0, Sv = Nv , Ev = Iv = 0. If we let y = Nh − Sh , z = Nv − Sv , we obtain the linearization y ′ = βv Iv − μh y Eh′ = βv Iv − (η + μh )Eh Ih′ = ηEh − (γ + μh )Ih z′ = −μv z + βh Ih Ev = βh Ih − (μv + ηv )Ev Iv′ = ηv Ev − μv Iv .

(6.4)

The corresponding characteristic equation is ⎤ ⎡ −λ 0 0 0 0 βv ⎥ ⎢ 0 0 0 βv ⎥ ⎢ 0 −(λ + ηh + μh ) ⎥ ⎢ ⎥ ⎢ 0 η −(λ + γ + μ ) 0 0 0 h h ⎥ = 0. det ⎢ ⎥ ⎢ 0 0 −βh −(λ + μv ) 0 0 ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ 0 0 −βh 0 −(λ + μv + ηv ) 0 0 0 0 0 ηv −(λ + μv )

Solutions of the linearization (6.4) are linear combinations of exponential functions whose exponents are the roots of the characteristic equation (the eigenvalues of the coefficient matrix of (6.4)). We can reduce this equation to a product of two factors and a fourth degree polynomial equation ⎤ 0 0 βv −(λ + ηh + μh ) ⎥ ⎢ ηh −(λ + γ + μh ) 0 0 ⎥ ⎢ λ(λ+μv )det ⎢ ⎥ = 0. ⎦ ⎣ 0 βh −(λ + μv + ηv ) 0 0 0 ηv −(λ + μv ) ⎡

The initial exponential growth rate is the largest root of this fourth degree equation, which reduces to g(λ) = (λ + ηh )(λ + γ + μh )(λ + μv + ηv )(λ + μv ) − βh βv ηh ηv = 0.

(6.5)

Since g(0) < 0 if R0 > 1, and since g(λ) is positive for large positive λ and g ′ (λ) > 0 for positive λ, there is a unique positive root of the equation g(λ) = 0, and this is the initial exponential growth rate. The initial exponential growth rate may be measured experimentally, and if the measured value is ρ, then from (6.5) we

234

6 Models for Diseases Transmitted by Vectors

obtain (ρ + ηh )(ρ + γ )(ρ + μv + ηv )(ρ + μv ) = βh βv ηh ηv = R0 ηh μv γ (μv + ηv ). From this, we obtain R0 =

(ρ + ηh )(ρ + γ )(ρ + μv + ηv )(ρ + μv ) . ηh μv γ (μv + ηv )

(6.6)

This gives a way to estimate the basic reproduction number from measurable quantities. In addition, the balance relation (6.1) allows us to calculate the values of βh and βv separately, which makes it possible to simulate the model. The same reasoning may be used for a short term epidemic model in which we ignore demographics in the host population. The results would be the same except that Λ(Nh ) and μh would be replaced by zero. By linearizing the system (6.2) at an equilibrium it is possible to show that the disease-free equilibrium is asymptotically stable if and only if R0 < 1 and that the endemic equilibrium exists only if R0 > 1 and is asymptotically stable.

6.3 Fast and Slow Dynamics In vector-borne disease transmission models in which the vector is an insect, the vector time scale is often much faster than the host time scale. In such models it is possible to consider two separate time scales [3]. We can make a quasi-steadystate hypothesis assuming that the vector population sizes remain almost constant [17]. We treat the vector population sizes as constants which depend on the host population sizes and approximate the system by a host model, which has fewer equations than the full system but may be more complicated in form. We describe the process in an SI R/SI epidemic model which is somewhat simpler than (6.2). Consider a host population with total population size Nh , and no demographics, and a vector population with birth and death rates μv and constant total population size Nv . We assume mass action contact rate with contact rates βh , βv , recovery rate γ for the host species and no recovery for the vector species. The resulting model is dSh Iv = −βh Sh dt Nv Iv dIh = βh Sh − γ Ih dt Nv

6.3 Fast and Slow Dynamics

235

Ih dSv = μv Nv − βv Sv − μv Sv dt Nh dIv Ih − μv Iv . = βv Sv dt Nh Because Sv + Iv is a constant Nv , we can reduce this model to a three-dimensional problem, dSh Iv = −βh Sh dt Nv Iv dIh = βh Sh − γ Ih dt Nv

(6.7)

dIv Ih = βv (Nv − Iv ) − μv Iv . dt Nh In order to obtain the basic reproduction number, we observe that a single infective host infects βh γ vectors in a wholly susceptible population of vectors, and each of these infects βv μv hosts in a wholly susceptible host population. Thus the number of secondary infections caused by an infective host (and also the number of secondary vector infections caused by an infective vector) is given by R0 =

βh βv . γ μv

However, one could also argue that the transition from host to vector to host should be viewed as two generations, and it may be more reasonable to say that R0 =

"

βh βv . γ μv

With either choice, the transition is at R0 = 1.

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6 Models for Diseases Transmitted by Vectors

It is possible to prove that there is an epidemic if and only if R0 > 1. A model for dengue fever of the form (6.7) has been studied in [15]. In terms of the above model, with time measured in days, this model has parameter values with μh much smaller than μv and βh much smaller than βv , expressing the fact that the vector dynamics are much faster than the host dynamics, and this suggests a model dSh Iv = −βh Sh dt Nv Iv dIh − αIh = βh Sh dt Nv ǫ

(6.8)

dIv Ih = βv (Nv − Iv ) − μv Iv , dt Nh

with ǫ a small positive constant. Other models describing two time scales may be found in [18, 19]. In fact, such a form is valid for all vector disease transmission models in which the vector population size is much greater than the host population size. We assume that Nh 1. In the simpler case when c = c∗ , the equilibrium value of I /N is given by  I∗ 1  κ 1 − , = N∗ κ + r2 + μ R0

(7.3)

which is a decreasing function of r1 and r2 as well. This shows that in the absence of the resistant strain, treatment is beneficial in reducing the disease burden. This is not the case when a resistant strain is considered, as shown in the two-strain model presented next.

7.2 A Two-Strain TB Model As mentioned earlier, treatment of active TB may take as long as 12 months, and lack of compliance with these treatments may lead to the development of antibiotic resistant TB. The one-strain model (7.1) can be extended to include both drug-sensitive (DS) and drug-resistant (DR) strains of TB, with possible development of resistance due to treatment failure. A transition diagram between the epidemiological classes is shown in Fig. 7.3. The latent and infectious individuals with sensitive TB are denoted by L1 and I1 , respectively. Two additional classes are included for the resistant strain, i.e., the latent and infectious classes with resistant strain denoted by L2 and I2 , respectively. The sensitive and resistant strains will be referred to as strain 1 and 2, respectively. Because it is very hard to cure a patient with resistant TB, we ignore the treatment of the resistant strain. Furthermore, assume that I2 individuals can infect S, L1 , and T individuals. The λ functions represent the forces of infection, which are given by λi (t) = ci

Ii , N

λ∗1 (t) = c1∗

I1 , N

i = 1, 2,

where c1 and c1∗ have the meaning as c and c∗ in the one-strain model (7.1); c2 is similar to c1 but for the resistant strain; ri (i = 1, 2) and μ are the same as in (7.1); κi denotes the progression rate from latent to infectious stage of strain i.

7.2 A Two-Strain TB Model

253

Fig. 7.3 A diagram for the two-strain TB model showing transitions between the epidemiological classes. The thicker arrow and the rate qr2 represent the development of resistant TB due to the failure of treatment for infections with the sensitive strain. The birth and death rates are omitted

The additional parameters are related to treatment failure and the development of resistant TB. For example, p + q denotes the proportion of those treated infectious individuals who did not complete their treatment, where the proportion p modifies the rate that departs from the sensitive TB latent class; qr2 I1 gives the rate at which individuals develop resistant TB because they did not complete the treatment of active TB. Therefore, p ≥ 0, q ≥ 0 and p + q ≤ 1. From the disease transmission diagram (see Fig. 7.3) we can write the following system of ordinary differential equations: I2 I1 S ′ = μN − c1 S − c2 S − μS, N N I1 I1 I2 L′1 = c1 S − (μ + κ1 )L1 − r1 L1 + pr2 I1 + c1∗ T − c2 L1 , N N N I1′ = κ1 L1 − μI1 − r2 I1 , I2 L′2 = qr2 I1 − (μ + κ2 )L2 + c2 (S + L1 + T ) , N I2 = κ2 L2 − μI2 , I1 I2 − c2 T − μT , T ′ = r1 L1 + (1 − p − q)r2 I1 − c1∗ T N N

(7.4)

where N = S + L1 + I1 + T + L2 + I2 is the total population, which remains constant. The detailed analysis of the model (7.4) is presented in [4]. System (7.4) has up to four possible equilibria denoted by E0 (infection-free), E1 (only the sensitive strain is present), E2 (only the resistant strain is present), and E ∗ (coexistence of both strains). The existence of these equilibria depends on the reproduction numbers for the sensitive and resistant strains, which are given by RS =



c1 + pr2 μ + r2



κ1 μ + κ1 + r1



(7.5)

254

7 Models for Tuberculosis

and RR =



c2 μ



κ2 μ + κ2



(7.6)

,

respectively. The dynamics of system (7.4) are dramatically different for the cases q = 0 and q > 0 (development of resistant TB due to treatment failure), particularly in terms of the number of equilibria and their stability, as well as the likelihood for coexistence of both strains. In addition to the reproduction numbers, there are two functions, RR = f (RS ) and RR = g(RS ), which divide the parameter region in the (RS , RR ) plane into sub-regions for the stability of equilibria: 1

f (RS ) =

1−RS 1 + (RS −AB)(1+1/B)    g(RS ) = C1 AB + C − 1 ± (AB + C − 1)2 + 4(RS − AB)C ,

(7.7)

for RS ≥ 1 where A=

pr2 , μ + κ1 + r1

B=

κ1 , μ + d1 + r2

C=

μ . μ + κ1 + r1

The properties of f and g include f (1) = g(1) = 1,

f (RS ) < g(RS ) for RS > 1

(see Fig. 7.4). The two curves of f and g and the lines Ri = 1 (i = S, R) divide the first quadrant of the (RS , RR ) plane into either four regions in the case q = 0 (see Fig. 7.4a) or three regions in the case q > 0 (see Fig. 7.4b). The stability results for system (7.4) in the case of q = 0 and q > 0 are summarized in Theorems 7.1 and 7.2, respectively. Theorem 7.1 Assume q = 0. Let Regions I –I V be as in Fig. 7.4a. (a) The disease-free equilibrium E0 is g.a.s. if (RS , RR ) is in Region I . (b) For RS > 1, the boundary equilibrium E1 is locally asymptotically stable if (RS , RR ) is in Region I I and unstable in Regions I I I and I V . (c) For RR > 1, the boundary equilibrium E2 is locally asymptotically stable if (RS , RR ) is in Region I V and unstable if in Regions I I and I I I . (d) The coexistence equilibrium E ∗ exists and is locally asymptotically stable if (RS , RR ) is in Region I I I . When q > 0, the equilibrium E1 (sensitive strain only) is never stable, and the coexistence region I I I is much larger than that in the case of q = 0, as stated in the following theorem and illustrated in Fig. 7.4b.

7.2 A Two-Strain TB Model

255

Fig. 7.4 (a) A bifurcation diagram for the system in the case q = 0. There are four Regions I , I I , I I I , and I V in the parameter space (RS , RR ). In Region I , E0 is a global attractor and other equilibria are unstable when they exist. In Regions I I and I V , E ∗ does not exist, while E1 and E2 are locally asymptotically stable, respectively. In Region I I I , E ∗ exists and is locally asymptotically stable . (b) A bifurcation diagram for the system in the case q > 0. There are three Regions I , I I I , and I V in the parameter space (RS , RR ) (E1 does not exist), in which E0 , E2 , and E ∗ are stable, respectively

Fig. 7.5 Phase portraits of solutions to (7.4) in the case of q = 0. The choice of parameter values gives a fixed value RS = 3.45. In (a) RR = 2 and (RS , RR ) ∈ IV. In (b) RR = 2.4 and (RS , RR ) ∈ III. In (c) RR = 1.2 and (RS , RR ) ∈ II. A circle indicates a stable equilibrium, and a triangle indicates an unstable equilibrium

Theorem 7.2 Assume that q > 0. Let Regions I –I I I be as in Fig. 7.4b. (a) The disease-free equilibrium E1 is g.a.s. if RS < 1 and RR < 1 (Region I ). (b) For RR > 1, the boundary equilibrium E2 is locally asymptotically stable if RS < 1 or if RS > 1 and RR > g(RS ) (Region I V ). E2 is unstable if RS > 1 and RR < g(RS ) (Region I I I ). (c) The equilibrium E3 exists and is locally asymptotically stable iff RS > 1 and RR < g(RS ) (Region I I I ). Figure 7.5 shows some simulation results of the model in the case of q = 0, illustrating the disease outcomes for (RS , RR ) in different regions as shown in Fig. 7.4a. In this figure, the parameter values used are μ = 0.143, c1 = 13, κ1 = 1, q = 0, p = 0.5, r1 = 1, r2 = 2, κ2 = 1. For this set of values, RS = 3.45.

256

7 Models for Tuberculosis

Figure 7.5a–c corresponds to different values of RR (or equivalently c2 ), for which (RS , RR ) is in Regions I V , I I I , and I I , respectively. These results demonstrate that lack of drug treatment compliance by TB patients may have an important implication for the maintenance of antibiotic resistant strains. To make the role of antibiotic resistance transparent, we first studied a special version of our two-strain model with two competing strains of TB: the typical strain plus a resistant strain that was not the result of antibiotic resistance (q = 0). In this last situation, we found that coexistence is possible but rare while later we noticed that coexistence is almost certain when the second strain is the result of antibiotic resistance. In our two-strain model there is a superinfection-like term c2 L1 I2 /N. Is this necessary to obtain the coexistence result because it is well known that superinfection can cause coexistence (see [16, 18])? The answer is no. In fact, it can be shown that in the absence of the superinfection-like term coexistence is still almost the rule when the second strain is the result of antibiotic resistance (see Fig. 7.4).

7.3 Optimal Treatment Strategies Analysis of the two-strain model in Sect. 7.2 demonstrated that treatment may facilitate the spread of resistant TB and increase the level of TB prevalence. Thus, the effort levels devoted to treating latent and infectious TB individuals may lead to different outcomes. Let u1 (t) and u2 (t) denote the time-dependent control efforts, which represent the fractions of individuals in L1 and I1 classes receiving prophylaxis and drug treatment, respectively, at time t. The state system with controls u1 (t) and u2 (t) reads: I2 I1 S ′ = μN − c1 S − c2 S − μS, N N I1 L′1 = c1 S − (μ + κ1 )L1 − u1 (t)r1 L1 N I2 I1 − c2 L1 , +(1 − u2 (t))pr2 I1 + c1∗ T N N I1′ = κ1 L1 − μI1 − r2 I1 ,

L′2 = (1 − u2 (t))qr2 I1 − (μ + κ2 )L2 + c2 (S + L1 + T ) I2 = κ2 L2 − μI2 ,

I2 , N



  I2 I1 − c2 T − μT , T ′ = u1 (t)r1 L1 + 1 − 1 − u2 (t) (p + q) r2 I1 − c1∗ T N N (7.8) with initial values S(0), L1 (0), I1 (0), L2 (0), I2 (0), T (0).

7.3 Optimal Treatment Strategies

257

The control functions, u1 (t) and u2 (t), are bounded, Lebesgue integrable functions. The “case-finding” control, u1 (t), represents the fraction of typical TB latent individuals that is identified and will be put under treatment (to reduce the number of individuals that may be infectious). The coefficient, 1 − u2 (t), represents the effort that prevents the failure of the treatment in the typical TB infectious individuals (to reduce the number of individuals developing resistant TB). When the “case-holding” control u2 (t) is near 1, there is low treatment failure and high implementation costs. Our objective function to be minimized is tf  B2 2 B1 2 u1 (t) + u (t) dt, L2 (t) + I2 (t) + J (u1 , u2 ) = 2 2 2

(7.9)

0

where we want to minimize the latent and infectious groups with resistant-strain TB while also keeping the cost of the treatments low. We assume that the costs of the treatments are nonlinear and take quadratic form here. The coefficients, B1 and B2 , are balancing cost factors due to size and importance of the three parts of the objective functional. We seek to find an optimal control pair, u∗1 and u∗2 , such that J (u∗1 , u∗2 ) = min J (u1 , u2 ), Ω

(7.10)

where Ω = {(u1 , u2 ) ∈ L1 (0, tf ) | ai ≤ ui ≤ bi , i = 1, 2} and ai and bi , i = 1, 2, are fixed positive constants. The necessary conditions that an optimal pair must satisfy come from Pontryagin’s maximum principle [19]. This principle converts (7.8)–(7.10) into a problem of minimizing pointwise a Hamiltonian, H , with respect to u1 and u2 : 6

H = L2 + I2 +

B1 2 B2 2 u + u + λi gi , 2 1 2 2

(7.11)

i=1

where gi is the right-hand side of the differential equation of the ith state variable. By applying Pontryagin’s maximum principle [19] and the existence result for the optimal control pairs from [13], we know that there exists an optimal control pair u∗1 , u∗2 and corresponding solution, S ∗ , L∗1 , I1∗ , L∗2 , I2∗ , and T ∗ , that minimizes J (u1 , u2 ) over Ω. Furthermore, there exist adjoint functions, λ1 (t), . . . , λ6 (t), such that I∗ I∗ I∗ I1∗ + c1∗ 2 + μ) + λ2 (−c1 1 ) + λ4 (−c1∗ 2 ), N N N N ∗ I∗ I λ′2 = λ2 (μ + κ1 + u1 (t)r1 + c1∗ 2 ) + λ3 (−κ1 ) + λ4 (−c1∗ 2 ) + λ6 (−u∗1 (t)r1 ), N N ∗ ∗ ∗ S S T λ′3 = λ1 (c1 ) + λ2 (−c1 − (1 − u∗2 (t))pr2 − c2 ) + λ3 (μ + r2 ) N N N λ′1 = λ1 (c1

258

7 Models for Tuberculosis

+ λ4 (−(1 − u∗2 (t))qr2 ) + λ6 (−(1 − (1 − u∗2 (t))(p + q))r2 + c2

T∗ ), N

λ′4 = −1 + λ4 (μ + κ2 ) + λ5 (−κ2 ),

L∗ S ∗ + L∗1 + T ∗ S∗ T∗ ) + λ2 (c1∗ 1 ) − λ4 (β ∗ ) + λ5 μ + λ6 (c1∗ ), N N N N ∗ ∗ ∗ ∗ I I I I (7.12) λ′6 = λ2 (−c2 1 ) + λ4 (−c1∗ 2 ) + λ6 (c2 1 + c1∗ 2 + μ), N N N N

λ′5 = −1 + λ1 (c1∗

with transversality conditions λi (tf ) = 0,

i = 1, . . . , 6

(7.13)

and N = S ∗ + L∗1 + I1∗ + L∗2 + I2∗ + T ∗ . Moreover, the characterization holds: 

u∗1 (t) = min max(a1 , B11 (λ2 − λ6 )r1 L∗1 ), b1 

u∗2 (t) = min max(a2 , B12 (λ2 p + λ4 q − λ6 (p + q)r2 I1∗ )), b2 .

(7.14)

Due to the a priori boundedness of the state and adjoint functions and the resulting Lipschitz structure of the ODEs, we obtain the uniqueness of the optimal control for small tf . The uniqueness of the optimal control follows from the uniqueness of the optimality system, which consists of (7.8) and (7.12)–(7.14). There is a restriction on the time interval in order to guarantee the uniqueness of the optimality system. This smallness restriction on the length on the time interval is due to the opposite time orientations of (7.8), (7.12), and (7.13); the state problem has initial values and the adjoint problem has final values. This restriction is very common in control problems (see [12, 15]). The optimal treatment is obtained by solving the optimality system, consisting of the state and adjoint equations. An iterative method is used for solving the optimality system. We start to solve the state equations with a guess for the controls over the simulated time using a forward fourth-order Runge–Kutta scheme. Because of the transversality conditions (7.13), the adjoint equations are solved by a backward fourth-order Runge–Kutta scheme using the current iteration solution of the state equations. Then, the controls are updated by using a convex combination of the previous controls and the value from the characterizations (7.14). This process is repeated and iteration is stopped if the values of unknowns at the previous iteration are very close to the ones at the present iteration. For the figures presented here, we assume that the weight factor B2 associated with control u2 is greater than or equal to B1 which is associated with control u1 . This assumption is based on the following facts: The cost associated with u1 will include the cost of screening and treatment programs, and the cost associated with u2 will include the cost of holding the patients in the hospital or sending people to watch the patients to finish their treatment. Treating an infectious TB individual takes longer (by several months) than treating a latent TB individual. In these three

7.3 Optimal Treatment Strategies

259

Fig. 7.6 The optimal control strategy for the case of B1 = 50 and B2 = 500

figures, the set of the weight factors, B1 = 50 and B2 = 500, is chosen to illustrate the optimal treatment strategy. Other epidemiological and numerical parameters are presented in [14]. In the top frame of Fig. 7.6, the controls, u1 (solid curve) and u2 (dashdot curve), are plotted as a function of time. In the bottom frame, the fractions of individuals infected with resistant TB, (L2 + I2 )/N, with control (solid curve) and without control (dashed curve) are plotted. Parameters N = 30,000 and β ∗ = 0.029 are chosen. Results for other parameters are presented in [14]. To minimize the total number of the latent and infectious individuals with resistant TB, L2 + I2 , the optimal control u2 is at the upper bound during almost 4.3 years and then u2 is decreasing to the lower bound, while the steadily decreasing value for u1 is applied over the most of the simulated time, 5 years. The total number of individuals L2 +I2 infected with resistant TB at the final time tf = 5 (years) is 1123 in the case with control and 4176 without control, and the total number of cases of resistant TB prevented at the end of the control program is 3053 (= 4176 − 1123). In Fig. 7.7, the controls, u1 and u2 , are plotted as a function of time for N = 6000, 12,000, and 30,000 in the top and bottom frame, respectively. Other parameters except the total number of individuals and c1∗ = 0.029 are fixed for

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7 Models for Tuberculosis

Fig. 7.7 The controls, μ1 and μ2 , are plotted as a function of time for N 6000, 12,000, and 30,000 in the top and bottom frame, respectively

=

these three cases. These results show that more effort should be devoted to “casefinding” control u1 if the population size is small, but “case-holding” control u2 will play a more significant role if the population size is big. Note that, in general, with B1 fixed, as B2 increases, the amount of u2 decreases. A similar result holds if B2 is fixed and B1 increases. In conclusion, our optimal control results show how a cost-effective combination of treatment efforts (case holding and case finding) may depend on the population size, cost of implementing treatments controls, and the parameters of the model. We have identified optimal control strategies for several scenarios. Control programs that follow these strategies can effectively reduce the number of latent and infectious resistant-strain TB cases.

7.4 Modeling of the Long and Variable Latency of TB As indicated in Figs. 7.1 and 7.2, the latent period of TB can range from a couple of years to lifetime. One of the approaches to incorporate this feature is to divide latent individuals into two classes based on the rates of progression to the disease stage,

7.4 Modeling of the Long and Variable Latency of TB

261

Fig. 7.8 A transition diagram of TB with fast and slow progressions [3]

with one class having a faster progression than the other. For example, based on the transition diagram shown in Fig. 7.8, Blower et al. considered the following model in [3]: S ′ = Λ − cSI − μS, L′ = (1 − p)cSI − (κ + r1 + μ)L, I ′ = pcSI + κL − (r2 + μ + d)I, C ′ = r1 L − μC, T ′ = r2 I − μT ,

(7.15)

which includes five epidemiological classes: susceptible (S), latently infected (L), infectious (I ), effectively chemoprophylaxed (C), and effectively treated (T ). The model assumes that a fraction p of the newly infected individuals will become infectious within 1 year (fast progression,p ≈ 0.05), while the remaining 1 − p fraction of newly infected individuals will enter the latent stage first and develop active TB at a rate κ (slow progression, approximately 1/20 (years)). The model suggests that for fast progression, an infected individual enters the infectious class I immediately. The parameters r1 and r2 denote the rates of prophylaxis and treatment, respectively. The per-capita natural and disease mortalities are μ and d, respectively. The control reproduction number of the model (7.15) is RC = RCfast + RCslow , where RCfast =

 cpΛ  μ

 1 r2 + μ + d

(7.16)

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7 Models for Tuberculosis

Fig. 7.9 Plots of the curve RC = 1 for severity levels of mild, moderate, and severe, which correspond to the basic reproduction number R0 values of 4, 9, and 17, respectively [3]

and RCslow =

 c(1 − p)Λ  μ

  κ 1 r1 + κ + μ r2 + μ + d

represent the reproduction numbers associated with the fast and slow paths, respectively. The formula (7.16) can be used to evaluate the possibility to eradicate tuberculosis either by treatment alone or by a combination of treatment and chemoprophylaxis, as demonstrated in Fig. 7.9 (adopted from [3]). To investigate the role of treatment failure for drug-sensitive (DS) cases in the prevalence of drug-resistant (DR) TB, Blower et al. [3] extended the one-strain TB model (7.15) to include a drug-resistant strain as follows:   S ′ = Λ − cS IS + cR IR S − μS,

L′S = (1 − p)cS IS S − (r1 + κ + μ)LS , IS′ = pcS IS S + κLS − (r2 + μ + d)IS , CS′ = r1 LS − μCS , TS′ = r2 (1 − q)IS − μTS , L′R = (1 − p)cR IR S − (κ + μ)LR , IR′ = pcR IR S + qr2 IS + κLR − (r2 δ + μ + d)IR , TR′ = δr2 TR − μTR ,

(7.17)

7.5 Backward Bifurcation in a TB Model with Reinfection

263

where the epidemiological classes associated with the DS and DR tuberculosis are indicated by the subscripts S and R, respectively. The parameter q represents the fraction of treatment failure for DS tuberculosis that leads to the development of DR tuberculosis, and the parameter δ denotes the relative effectiveness of treatment for a resistant case. Blower et al. [3] showed that the model (7.17) can help gain insights into the impact of treatment failure (q) on the development of DR tuberculosis. Another approach to incorporate the long and variable latency is to consider an arbitrarily distributed latency, in which case the model consists of a system of integro-differential equations. Examples can be found in [10, 11]. It is shown in [10] that the model with a distributed latency has the same dynamical behavior as the ODE model, although it can provide a more detailed description for the reproduction number as it involves a more realistic distribution of the latent period. The model considered in [11] includes both DS and DR strains with infection age-dependent progression (e.g., the progression distribution shown in Fig. 7.2).

7.5 Backward Bifurcation in a TB Model with Reinfection As mentioned earlier, the latent period of TB can be as long as many years and even lifetime. Re-exposure to TB bacilli through repeated contacts with individuals with active TB may accelerate the progression of LTBI towards active TB, and exogenous reinfection (i.e., acquiring a new infection from another infected individual) may occur. To investigate the impact of exogenous reinfection in the spread and control of TB, we can extend the one-strain model (7.1) by incorporating the exogenous reinfection. It is demonstrated in [9] that exogenous reinfection may play a fundamental role in the transmission dynamics and the epidemiology of TB at the population level. Particularly, the model is capable of exhibiting a backward bifurcation, i.e., a stable endemic equilibrium can exist even when the reproduction number is less than 1. Although some studies (e.g., see [23]) find that, for parameter values in a certain range, the onset of the backward bifurcation is unlikely to occur, other scenarios are possible in which the conditions for backward bifurcation can be satisfied. In either case, it is helpful to know that exogenous reinfection may play an important role in TB dynamics, which can be critical for the design of control programs for TB. An extension of the one-strain model (7.1) when reinfection is included takes the form: I S ′ = μN − cS − μS, N I I I L′ = cS + c∗ T − pcL − (κ + μ)L, N N N I I ′ = pcL + κL − (r + μ)I, N I T ′ = rI − c∗ T − μT . N

(7.18)

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7 Models for Tuberculosis

The model (7.18) ignores the treatment of latent individuals and only infectious individuals may receive treatment at the rate r. The parameter p is a factor reflecting the difference between primary infections (infection from susceptibles) and exogenous reinfections (infection from latent individuals). A value of p ∈ (0, 1) implies that a primary infection provides some degree of cross-immunity to exogenous reinfections. Other parameters have the same meaning as in (7.1). The reproduction number for (7.18) is R0 =

cκ . (κ + μ)(r + μ)

(7.19)

The usual result that the disease-free equilibrium is locally asymptotically stable when R0 < 1 still holds. However, the usual result that there is no endemic equilibrium when R0 < 1 no longer holds. To show that an endemic equilibrium may exist when R0 < 1, consider the simpler case when c∗ = c. Let U ∗ = (S ∗ , L∗ , I ∗ , T ∗ ) denote an endemic equilibrium, i.e., I ∗ > 0. Let x = I ∗ /N, then μ S∗ = , N μ + cx ∗

(μ + r)x ∗ L∗ = , N κ + pcx ∗

rx T∗ = , N μ + cx

and x ∗ is a solution of the quadratic equation Ax 2 + Bx + C = 0,

(7.20)

where A = pR0 ,

B = (1 + p + Q)DE − pR0 ,

C = DE Q

 1  −1 , R0

and DE = κ/(μ+κ) and Q = κ/(μ+r). Note that DE < 1 denotes the probability that a latent individual survives and becomes infective. Let Rp =

  1 DE (1 + p − Q) + s DE Q(p − pDE − DE ) p

(7.21)

and p0 =

(1 + Q)DE . 1 − DE

(7.22)

Then Rp < 1 if p > p0 , and B 2 − 4AC > (=, (=, p0 ). The number of infectious individuals I (t) is plotted. It illustrates that, depending on the initial conditions, the solution will converge to either the infection-free equilibrium (the solid curves) or the stable positive equilibrium (dashed curves)

Theorem 7.3 Let R0 , Rp , and p0 be as defined in (7.19), (7.21), and (7.22). (a) If R0 > 1, then system (7.18) has exactly one endemic equilibrium and it is locally asymptotically stable (b) If R0 < 1, then the disease-free equilibrium is locally asymptotically stable Moreover, (i) for each p > p0 there exists a positive constant Rp < 1 such that system (7.18) has exactly two (one, none) positive equilibria when R0 > ( =, 5% prevalence of MDR-TB). The findings of the model suggest that the levels of MDR are driven by case-finding rates, cure rates, and amplification probabilities (the probability that a case will develop further resistance during treatment). In [7], heterogeneity in the relative fitness of MDR strains is incorporated in a TB model. The model includes two resistant strains (one is more fit than the other), as well as a drug-sensitive strain, and is used to study the impact of initial fitness estimates on the emergence of MDR-TB. The model results show that “even when the average relative fitness of MDR strains is low and a well-functioning control program is in place, a small sub-population of a relatively fit MDR strain may eventually outcompete both the drug-sensitive strains and the less fit MDR strains.” A two-strain TB model with reinfection is considered in [21] to study the role of reinfection in the transmission dynamics of drug-resistant TB and the coexistence of sensitive and resistant strains of TB. In [17] multiple sub-populations are considered with one sub-population being genetically susceptible to TB. Different strategies involving treatment of latent TB infections and active TB disease are examined and in different populations are examined. Results from the model analysis suggest that the presence of a genetically susceptible sub-population dramatically alters effects of treatment. To study the impact of treatment failure and its influence in the criteria for the control of drug-resistant TB, a model with multiple stages of treatment failure with different probabilities of leading to resistant TB is considered in [8]. Model results indicate that case detection and treatment can be a critical factor in the control of MDR-TB. To explore the influence of HIV on TB prevalence, models including the interaction between TB and HIV can be used, in which case the model analysis can be very challenging due to the possible coinfections of TB and HIV. For example, a model incorporating both TB and HIV with multiple stages of HIV is studied in [20]. The model results suggest that “an HIV epidemic can significantly increase the

7.7 Project: Some Calculations for the Two-Strain Model

267

frequency and severity of tuberculosis outbreaks, but that this amplification effect of HIV on tuberculosis outbreaks is very sensitive to the tuberculosis treatment rate.” Another model that includes both TB and HIV is studied in [22]. The model results suggest that the accelerated progression from LTBI to active TB in individuals coinfected with HIV can have a significant influence in TB prevalence.

7.7 Project: Some Calculations for the Two-Strain Model Exercise 1 Derive the formulas of the reproduction numbers RS and RR given in (7.5) and (7.6), respectively, for the two-strain TB model (7.4) by considering the stability of the disease-free equilibrium

 (0) (0) (0) (0) E0 = S (0) , L1 , I1 , L2 , I2 , T (0) = (N, 0, 0, 0, 0, 0).

The Jacobian a E0 is ⎡

−μ 0 −c1 0 −c2 ⎢ 0 −(μ + κ + r ) c + pr 0 0 1 1 1 2 ⎢ ⎢ κ1 −(μ + r2 ) 0 0 ⎢ 0 J (E0 ) = ⎢ ⎢ 0 −(μ + κ2 ) c2 0 qr2 ⎢ ⎣ 0 −μ 0 0 κ2 0 r1 (p + q)r2 0 0

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥. 0 ⎥ ⎥ 0 ⎦ −μ

Clearly, J (E0 ) has the negative eigenvalues μ with multiplicity 2, and other eigenvalues are given by the eigenvalues of J1 and J2 , where   −(μ + κ1 + r1 ) c1 + pr2 , J1 = κ1 −(μ + r2 )

  −(μ + κ2 ) c2 J2 = . κ2 −μ

Show that J1 has negative eigenvalues if and only if RS < 1 and J2 has negative eigenvalues if and only if RR < 1. Exercise 2 Consider the two-strain TB model (7.4) and conduct numerical simulations of the system for parameter values specified below. Given the parameter values μ = 0.0143 (or 1/μ = 70), c1 = 13, κ1 = κ2 = 1, r1 = 1, r2 = 2, q = 0, p = 0.5. Choose the values of c2 such that (a) E1 is stable; (b) E2 is stable; (c) E ∗ is stable.

268

7 Models for Tuberculosis

Exercise 3 Similar to Exercise 2 but for the case when q > 0. Given the same values for other parameters as in Exercise 2, and choose the values of c2 such that (a) E2 is stable; (b) E ∗ is stable.

7.8 Project: Refinements of the One-Strain Model The model (7.1) is a one-strain model for TB, which assumes that infected people stay in the latent stage L before entering the infective stage I . The model (7.15) includes fast and slow progression (represented by p and 1 − p, respectively), which assumes that with fractions 1 − p and p infected individuals will enter the latent L and infectious I stages, respectively. In both models, the stage duration is assumed to be exponentially distributed with parameter k. That is, the probability that an infected individual has not become infective s units of time after infection is e−ks . This assumption may not be appropriate for TB due to the long and variable latency, as shown in Figs. 7.1 and 7.2. To examine whether or not a model with a more realistic assumption on the distribution of latent period, we can consider a model in which the exponential survival function e−ks is replaced by a general distribution p(s), as described below. Let p(s) be a function representing the proportion of those individuals latent at time t and who, if alive, are still infected (but not infectious) at time t + s. Then −p(τ ˙ ) is the rate of removal of individuals from E class into I class τ units of time after becoming latent. Assume that p(s) ≥ 0,

p(s) ˙ ≤ 0,

p(0) = 1,



0

∞

p(s)ds < ∞.

Let S(t), E(t), I (t), and T (t) denote the number of individuals in the susceptible, latent, infectious, and treated classes, respectively. Consider the following model with p(s) being the arbitrary distribution of the latent stage: S ′ = Λ − cS NI − μS, t  I (s) p(t − s)e−(μ+r1 )(t−s) ds, E(t) = E0 (t) + cS(s) + c∗ T (s) N (s)  t τ 0   I (s) −(μ+r )(τ −s) 1 e I (t) = cS(s) + c∗ T (s) N (s) 0 0 ×[−p(τ ˙ − s)e−(μ+r2 )(t−τ ) ]ds dτ + I0 e−(μ+r2 )t + I0 (t), ′ T = r1 E + r2 I − c∗ T NI − μT , N = S + E + I,

(7.23)

7.9 Project: Refinements of the Two-Strain Model

269

where E0 (t) denotes those individuals in E class at time t = 0 and still in the latent class, I0 (t) denotes those initially in class E who have moved into class I and are still alive at time t, and I0 e−(μ+r2 )t with I0 = I (0) represents those who are infective at time 0 and are still alive and in I class. E0 (t) and I0 (t) are assumed to have compact support (that is, they vanish for large enough t). All other parameters have the same meanings as in model (7.1). Question 1 Derive the formula below for the reproduction number R0 R0 = c



∞

a(τ )dτ,

(7.24)

0

where a(u) is defined by a(t − s) =



s

t

e−(μ+r1 )(τ −s) [−p(τ ˙ − s)e−(μ+r2 )(t−τ ) ]dτ.

Provide biological interpretations for the a(u) and the factors in the expression of a(u). Question 2 Show that the disease will die out if R0 < 1 and persist if R0 > 1. Question 3 Find all biologically feasible steady-state solutions of the model (7.23). Determine the condition for the existence of a coexistence steady state (i.e., I > 0 and J > 0). Question 4 For each steady state, identify the conditions under which the steady state is locally asymptotically stable. Question 5 Does the model with a more realistic distribution for the latent stage provide new qualitative behavior of the disease dynamics in comparison with the ODE model (7.1)? Identify the additional insights that the model (7.23) can provide.

7.9 Project: Refinements of the Two-Strain Model Model (7.4) is a two-strain model for TB with drug-sensitive and drug-resistant strains. In this model, the latent stages for both strain are assumed to be exponentially distributed with parameters κ1 and κ2 . The long and variable latency for the sensitive strain make this assumption unrealistic (the latent period for the resistant strain is much shorter). The model below is an alternative two-strain model with distributed delay in latency, in which p(θ ) is used to describe the progression from latent stage to infectious stage and θ is the age of infection (time since becoming infected):

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7 Models for Tuberculosis

dS = μN − (μ + λ1 (t) + λ2 (t)) S + (1 − r)χ dt



∞

p(θ )i(θ, t)dθ,

0

∂i ∂i + + μ i(θ, t) + (1 − r + qr)χp(θ )i(θ, t) = 0, ∂t ∂θ  ∞ dJ p(θ )i(θ, t)dθ, = λ2 (t)S − μJ + qrχ dt 0

(7.25)

i(0, t) = λ1 (t)S(t), S(0) = S0 > 0, i(θ, 0) = i0 (θ ) ≥ 0, J (0) = J0 > 0. S(t) is the number of susceptibles at time t; i(θ, t) denotes the density of infection age θ individuals with the drug-sensitive strain at time t; J (t) is the number of infected individuals with a drug-resistant strain at time t; and N = S + I + J , where I (t) =



∞

i(θ, t)dθ

0

is the total number of individuals infected with the sensitive strain. The function p(θ ) (0 ≤ p(θ ) ≤ 1) is assumed constant in time and is based on experimental data (e.g., the distribution in Fig. 7.2). Note that p(θ )i(θ, t) represents the age density of infectious individuals. The transmission rates of sensitive and resistant strains are  ∞ c1 J (t) , (7.26) λ1 (t) = p(θ )i(θ, t)dθ and λ2 (t) = c2 N (t) 0 N (t) respectively, with c1 and c2 being the per-capita transmission rates of the sensitive and resistant strains. The rate at which sensitive-strain-infected individuals leave the i class due to treatment is (1 − r + qr)χp(θ ), where χ denotes the treatment rate for individuals with drug-sensitive TB. The factor (1 − r + qr) introduces the effect of incomplete treatment: a fraction r of the treated individuals with sensitive TB do not recover due to incomplete treatment, and the remaining fraction 1 − r is actually cured and becomes susceptible again. Moreover, among the individuals who do not finish their treatment, a fraction q of them will develop drug-resistant TB and the remaining fraction will remain latent. The per-capita birth and natural death rates are assumed to be the same and equal to μ. In this model, for simplicity, treated individuals are assumed to have the same susceptibility. Question 1 Let v(t) = i(0, t) = λ1 (t)S(t). Show that system (7.25) is equivalent to the following system, which is easier to analyze:

References

v(t) =

271

N(t) − J (t) −



0

t

K0 (θ )v(t − θ )dθ 

N (t) 



dJ = β2 N(t) − J (t) − dt  −mJ (t) +

0

t

0 t

t

K1 (θ )v(t − θ )dθ + F˜1 (t),  J (t) K0 (θ )v(t − θ )dθ N (t) 0

K2 (θ )v(t − θ )dθ + F˜2 (t),

dN = b(N )N (t) − μN(t) − δJ (t), dt

(7.27)

where F˜i (t) involve parameters and initial condition with limt→∞ F˜i (t) = 0, i = 1, 2, and

K0 (θ ) = e

−μθ−



θ

(1 − r + qr)χp(s)ds

,   d β1 K0 (θ ) + μK0 (θ ) , K1 (θ ) = β1 p(θ )K0 (θ ) = − χ (1 − r + qr)  dθ  d qr K0 (θ ) + μK0 (θ ) . K2 (θ ) = qrχp(θ )K0 (θ ) = − (1 − r + qr) dθ 0

(7.28)

Question 2 Let R1 and R2 denote the reproduction numbers for the sensitive and resistant strains. Derive a formula for R1 and R2 . Question 3 Explore how the existence and stability of steady states of the model (7.25) depend on R1 and R2 . Question 4 Does the model with a more realistic distribution for the latent stage for the sensitive strain provide new qualitative behavior of the disease dynamics in comparison with the ODEs model (7.4)? Identify the additional insights that the model (7.25) may provide.

References 1. Aparicio, J. P., and C. Castillo-Chavez (2009) Mathematical modelling of tuberculosis epidemics, Math. Biosc. Eng. 6: 209–237. 2. Blower, S. M., and T. Chou (2004) Modeling the emergence of the ‘hot zones’: tuberculosis and the amplification dynamics of drug resistance, Nature Medicine 10: 1111–1116. 3. Blower, S. M., P.M. Small, and P.C. Hopewell (1996) Control strategies for tuberculosis epidemics: new models for old problems, Science 273: 497–500. 4. Castillo-Chavez, C., and Z. Feng (1997) To treat or not to treat: the case of tuberculosis. J. Math. Biol. 35: 629–656. 5. Castillo-Chavez, C., and Z. Feng (1998) Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math.Biosc. 151: 135–154. 6. CDC, Tuberculosis treatment (2014) http://www.cdc.gov/tb/topic/treatment/.

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7. Cohen, T., and M. Murray (2004) Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness Nature Medicine 10: 1117–1121. 8. Dye, C., and B.G. Williams (2000) Criteria for the control of drug-resistant tuberculosis, Proc. Nat. Acad. Sci. 97: 8180–8185. 9. Feng, Z., C. Castillo-Chavez, and A.F. Capurro (2000) A model for tuberculosis with exogenous reinfection, Theor. Pop. Biol. 57: 235–247. 10. Feng, Z., W. Huang, and C. Castillo-Chavez (2001) On the role of variable latent periods in mathematical models for tuberculosis, J. Dyn. Diff. Eq. 13: 435–452. 11. Feng, Z., M. Iannelli, and F.A. Milner (2002) A two-strain tuberculosis model with age of infection, SIAM J. Appl. Math. 62: 1634–1656. 12. Fister, K. R., S. Lenhart, and J.S. McNally (1998) Optimizing chemotherapy in an HIV model, Electronic J. Diff. Eq. 32: 1–12. 13. Fleming, W., and R. Rishel (1975) Deterministic and Stochastic Optimal Control, Springer. 14. Jung, E., S. Lenhart, and Z. Feng (2002) Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems Series B 2: 473–482. 15. Kirschner, D., S. Lenhart, and S. Serbin (1997) Optimal control of the chemotherapy of HIV J. Math. Biol. 35: 775–792. 16. Levin, S. and D. Pimentel (1981) Selection of intermediate rates of increase in parasite-host systems, Am. Naturalist 117: 308–315. 17. Murphy, B. M., B.H. Singer, and D. Kirschner (2003) On treatment of tuberculosis in heterogeneous populations, J. Theor. Biol. 223: 391–404. 18. Nowak, M.A., and R.M. May (1994) Superinfection and the evolution of parasite virulence, Proc. Roy. Soc. London, Series B: Biological Sciences 255: 81–89. 19. Pontryagin, L.S. (1987) Mathematical Theory of Optimal Processes, CRC Press. 20. Porco, T.C., P.M. Small, and S.M. Blower (2001) Amplification dynamics: predicting the effect of HIV on tuberculosis outbreaks, J. Acquired Immune Deficiency Syndromes 28: 437–444. 21. Rodrigues, P., M.G.M. Gomes, and C. Rebelo (2007) Drug resistance in tuberculosis, a reinfection model, Theor. Pop. Biol. 71: 196–212. 22. Roeger, L-I.W., Z. Feng, and C. Castillo-Chavez (2009) Modeling TB and HIV co-infections Math. Biosc. Eng. 6: 815–837. 23. Singer, B.H., and D.E. Kirschner (2004) Influence of backward bifurcation on interpretation of R0 in a model of epidemic tuberculosis with reinfection, Math. Biosc. Eng. 1: 81–93. 24. Styblo, K. (1991) Selected papers. vol. 24, Epidemiology of tuberculosis, Hague, The Netherlands: Royal Netherlands Tuberculosis Association. 25. WHO. Global tuberculosis report 2013 http://www.who.int/tb/publications/global_report/.

Chapter 8

Models for HIV/AIDS

8.1 Introduction Acquired immunodeficiency syndrome (AIDS) was first identified as a new disease in the homosexual community in San Francisco in 1981. The human immunodeficiency virus (HIV) was identified as the causative agent for AIDS in 1983. The disease has several very unusual aspects. After the initial infection, there are symptoms, including headaches and fever for 2 or 3 weeks. Transmissibility is high for about 2 months, and then there is a very long latent period during which transmissibility is low. At the end of this latent period, which may last 10 years, transmissibility rises, signaling the development of full-blown AIDS. In the absence of treatment, AIDS is invariably fatal. Now, HIV can be treated with a combination of highly active antiretroviral therapy (HAART) drugs, which both reduce the symptoms and prolong the period of low infectivity. While there is still no cure for AIDS, treatment has made it no longer a necessarily fatal disease. To describe the variation of infectivity for HIV, one possibility would be to use a staged progression model, with multiple infective stages having different infectivity. Another possibility would be to use an age of infection model. HIV is transmitted in many ways, the most common of which are sexual contact, either heterosexual or homosexual, shared drug injection needles, and contaminated blood transfusions. Vertical transmission from mother to child is also possible. In the past, transfusions of contaminated blood were another source of disease transmission, but in developed countries screening of blood since 1985 has eliminated blood transfusions as a transmission mode. A full model for HIV/AIDS should include a variety of transmission modes, and might take into account of many factors including the level of sexual activity, drug use, condom use, and the sexual contact network, resulting in large scale systems with many parameters that need to be estimated from data. Models were developed first for homosexual transmission. In this chapter, we will consider not only models for disease transmission in a homosexual community (the current terminology is © Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_8

273

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8 Models for HIV/AIDS

men having sex with men, or MSM), but also models that include heterosexual transmission through female sex workers. We also consider modes that include the joint disease dynamics of HIV and TB and the synergy between HIV and HSV-2. The identification of the human immunodeficiency virus [11, 55, 56, 58, 107] captured the attention of theoreticians and modelers as AIDS became one of the most feared diseases nearly three decades ago. Most of the initial modeling contributions focused on the study of the transmission dynamics of HIV at the population level since little was known about the epidemiology of HIV and, as expected, modeling was carried out first under simple settings and crude assumptions [3– 7, 10, 19, 26, 32–35, 46, 48, 49, 57, 59, 65, 67–71, 75, 76, 79, 81, 88, 95–97, 102, 103]. An overview of the “state of the art” on the transmission dynamics of HIV modeling in the 1980’s is found in [30], the review papers [97, 99], or in the books [8, 30, 63]. The modeling studies in [32–35, 67, 102, 103] focused on the impact that changes in the pool of susceptibles, disease-induced mortality, heterogeneous mixing, vertical transmission, asymptomatic carriers, variable infectivity, and incubation (or latent) and infective periods may have on the dynamics of sexually transmitted HIV. Efforts to model the risk of infection from sexual partner selection or from within and between group mixing became central to the research of various groups studying HIV dynamics. Other studies focused on the role of gender, core populations, and heterogenous mixing contact rates on HIV dynamics. These were naturally involved in the development of sexual-behavior surveys and data collection on sexual and “dating” activity, as well as on the mathematical modeling and analysis of heterogenous “mixing” frameworks (see [20, 21, 23, 24, 27, 28, 36– 39, 41, 47, 78, 93]). The overview in [83] highlights the potential role of sexual activity and drinking on the dynamics of STDs [47, 65, 66, 93] and while the adaptive dynamics generated by changing behaviors in response, to a multitude of factors, were rarely explored, some earlier attempts were also carried out as a result of the HIV pandemic [22, 60]. As described in these historical papers [3–5], knowledge of these periods was quickly identified as critical to the initial efforts to predict the dynamics of HIV. In [35] it is observed that: “The duration of the latent period is thought to be a few days to a few weeks [3–5], and while the duration of the infectious period is not yet known, those individuals that develop full-blown AIDS have an average incubation period estimated variously at 35–47 months [88], 66 months [3], and as high as 96 months [81].” This estimate is continually being revised as information and experience accumulate. However, even the most conservative estimate suggests that it may be reasonable to approximate the infectious period by the incubation period; that is, to assume a negligible latent period. Pickering et al. [88] stress that the ability to transmit HIV is not constant, as individuals are most infectious 3–16 months following exposure, and recent studies [58, 77, 94] report the existence of two peaks of infectiousness, one taking place a few weeks after exposure and the other before the onset of “full-blown” AIDS. In the context of the dynamics of a homosexually-active homogeneously-mixing population, the reproduction number is given by R0 = λC(T )D, where λ denotes the probability of transmission per partner, C(T ) the mean number of sexual

8.2 A Model with Exponential Waiting Times

275

partners an average individual has per unit time when the population density is T , and D the death-adjusted mean infective period (see [35]). Since HIV is a slow disease, if R0 ≤ 1 it will die out while if R0 ≥ 1 it will persist in the presence of a small number of infected/infective individuals. The mathematical analysis and numerical simulations in [35] suggest that whenever the incubation period distribution is exponential the reproduction number R0 is a global bifurcation parameter (transcritical bifurcation), that is, as R0 crosses 1 a global transfer of stability from the disease-free state to the endemic equilibrium takes place, and vice versa. Local results do not depend on the distribution of times spent in the infective categories (the survivorship functions). Keeping a suite of parameters fixed [35] allowed for the comparison of the exponential incubation period distribution versus a piecewise constant survivorship (individuals remain infective for a fixed length of time). It was found that for “. . . some realistic parameters we can see (at least in these cases) that the reproduction numbers corresponding to these two extreme cases do not differ by more than 18% whenever the two distributions have the same mean [35].” The inclusion of heterogeneity via the introduction of a large number of subgroups limited the forecasting capability of these models due to factors that included increased levels of uncertainty (more parameters). The use of multigroup models raised the expected modeling and parameter estimation challenges [20, 21, 23, 24, 27, 28, 36–38, 41, 65, 66, 93]. In addition, the analyses of some of these models generated novel dynamic behavior, questioning, possibly for the first time in epidemiology, the centrality of the role of the basic reproduction number in the identification and development of control, or education, or intervention measures. For example, the natural asymmetry present in disease transmission as a result of prevalent alternative modes of sexual engagement proved to be capable of giving rise to the existence of multiple equilibria [33, 34, 67]; an unexpected outcome at that time.

8.2 A Model with Exponential Waiting Times A single homosexually-active population is divided into three classes. S denoting the number of susceptible individuals, I infective individuals, and A former I individuals who have developed full-blown AIDS (see Fig. 8.9). We assume that all HIV-infected individuals will eventually develop full-blown AIDS (unless they die first from other causes). This, unfortunately, may be the most realistic as evidence accumulates that AIDS is a progressive disease. Later, we will suggest a project to develop a model under the assumption that some fraction of infected individuals will escape progression to full-blown AIDS. Originally, a latent class (i.e., those exposed individuals that are not yet infectious) was not included because it was believed then that the time spent in that class is short. It is further assumed that individuals who develop full-blown AIDS are no longer actively infective, that is, that they have no sexual contacts; it is also assumed that infected individuals become

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infective immediately. Finally, it is assumed that infective individuals acquire AIDS at the constant rate αI per unit time and become sexually inactive at the constant rate α per unit time. Therefore, 1/(μ + αI ) gives the mean incubation period and 1/(μ + α) gives the mean sexual life expectancy. The introduction of the model requires additional definitions. Λ will denote the constant recruitment rate into the susceptible class (individuals who are sexually active); μ the constant per-capita natural mortality rate; d the per-capita constant disease-induced mortality due to AIDS. The function C(T ) models the mean number of sexual partners an average individual has per unit time when the population density is T ; λ (a constant) denotes the average sexual risk per infected partner; λ is often thought as the product iφ [68], where φ is the average number of contacts per sexual partner and i the conditional probability of infection from a sexual contact when the latter is infected. Kingsley et al. [72] had presented (not surprising) evidence that the probability of seroconversion (infection) increases with the number of infected sexual partners. Hence, λC(T ) models the transmission rate per unit time per infected partner when the size of the sexually active population is T . Using the modeling framework published in [3, 4] with the help of Fig. 8.1, we arrive at the following epidemiological model [35] for sexually transmitted HIV under the assumption of exponential waiting times in the infection classes. dS(t) S(t)I (t) = Λ − λC(T (t)) − μS(t) dt T (t) S(t)I (t) dI (t) = λC(T (t)) − (αI + μ)I (t) dt T (t)

(8.1)

dA(t) = αI I (t) − (α + μ)A(t) dt where T = I + S.

(8.2)

The fraction I /T can be thought of as representing the fraction of contacts that a susceptible individual has with a randomly selected infective individual. Here λC(T )SI /T denotes the number of newly-infected individuals per unit time since

Λ

λC(T )S

S µ

I T

d αI

I

A

µ

µ

Fig. 8.1 Flow diagram: single group model in the case when all infected people will progress to AIDS

8.2 A Model with Exponential Waiting Times

277

individuals in classes A are sexually inactive. A plausible assumption for modeling C(T ) would be to assume that it is approximately linear for small T approaching a saturation level for large values of T [62]. Here, it is assumed that C(T ) is a differentiable and increasing function of T (except when noted). Anderson et. al. [4] observe that C(T ), the mean number of sexual partners per unit time, underestimates the importance of highly active individuals and that consequently, modifications should be made to this framework in order to properly account for their role. The analysis of the system (8.1) found in [35] makes the following assumptions concerning C(T ): C ′ (T ) ≥ 0,

C(T ) > 0,

(8.3)

with prime denoting the derivative with respect to T . The dynamics of S and I are independent of A (by construction). The system is well-posed, that is, if S(0) ≥ 0, I (0) ≥ 0, A(0) ≥ 0 then a unique solution exists with S(t) ≥ 0, I (t) ≥ 0, A(t) ≥ 0 for t ≥ 0. As it is the case with most of the epidemiological systems addressed in this book, system (8.1) always has the disease-free equilibrium given by (S, I, A) =



 Λ , 0, 0 , μ

(8.4)

and under certain assumptions it also supports a unique endemic equilibrium. The stability of the disease-free equilibrium (8.4) is determined by the nondimensional ratio     Λ 1 C , (8.5) R0 = λ σI μ that is, by the basic reproduction number. In the definition of R0 , σI = αI + μ, and R0 denotes the number of secondary infections generated by a single infective individual in a population of susceptibles at a demographic steady state. R0 is given by the product of the three factors (epidemiological parameters): λ (the probability of transmission per partner), C(Λ/μ) (the mean number of sexual partners that an average susceptible individual has per unit time when everybody in the population is susceptible), and D=



1 σI



.

(8.6)

The death-adjusted mean infective period is D = DI with DI denoting the deathadjusted mean infectious period 1/σI of the I class. The use of the dimensionless ratio, R0 = λC(Λ/μ)D leads to the following result [35]: Theorem 8.1 If R0 < 1 then the equilibrium (Λ/μ, 0, 0) of the system (8.1) is globally asymptotically stable.

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That is, every solution of (8.1) (S(t), I (t), A(t)) with S(0) ≥ 0, I (0) ≥ 0, A(0) ≥ 0 tends to (Λ/μ, 0, 0) as t → +∞. That is, the condition R0 ≤ 1 is sufficient to guarantee that the disease will eventually die out in this population. An endemic equilibrium (S ∗ , I ∗ , A∗ ) of (8.1) satisfies Λ=



 Λ − μS ∗ − μ S∗, σI

I∗ =

Λ − μS ∗ , αI + μ

A∗ =

αI I ∗. αI + μ

In [35] it has also been established (following some of the same arguments used in other chapters) that: Theorem 8.2 If R0 > 1, there is a unique endemic equilibrium (S ∗ , I ∗ , A∗ ), which is locally asymptotically stable, and the disease-free state (Λ/μ, 0, 0) is unstable. We can summarize the situation (full details of all proofs are in [35]) as follows: The disease-free state of system (8.1) is globally asymptotically stable when R0 > 1 and unstable if R0 > 1. When R0 > 1, this system has a unique locally asymptotically stable endemic equilibrium. There is a transfer of stability to the endemic state as R0 crosses one. Further, when R0 > 1 it was shown, as one would expect, that the endemic equilibrium is also globally asymptotically stable. The reproduction number (R0 ) increases proportionately with the transmission probability and the average number of sexual partners; it may also increase in proportion to the rate of recruitment of individuals to the susceptible class through C(T ). R0 is an increasing function of the mean infective period D and may be a decreasing function of the mortality rate (depending on the functional expression for C(T )).

8.3 An HIV Model with Arbitrary Incubation Period Distributions The use of exponential waiting distributions in the I class corresponds to the requirement that the per-capita removal rate from the I class (by the development of full-blown AIDS symptoms) into the A class is constant. It would be clearly an improvement in the model of Sect. 8.2 if we were to move from constant to variable removal rates and this is what we do in this section (the ideas follow those in [19, 35]). Hence, it is still assumed that individuals become immediately infective (that is, we continue to neglect the latent period) and continue to divide the at risk population into the three classes: S, I , and A. The parameters λ = iφ, Λ, μ, d, and p have the same meaning as in Sect. 8.2; however, the removal rates are modified through the introduction of the function PI (s) representing the proportion of individuals who become I −infective at time t and that, if alive, are still infective

8.3 An HIV Model with Arbitrary Incubation Period Distributions

279

at time t + s (survive as infective). The survivorship function PI is non-negative and non-increasing, and PI (0) = 1. It is further assumed that  ∞ PI (s)ds < ∞, 0

and thus, −P˙I (x) is the rate of removal of individuals from the class I into the class A, x time units after infection. The number of new infections occurring at time x is λC(T (x)S(x)I (x)/T (x) where we have kept the meaning of C(T ), I , and T as in Sect. 8.2. The rate of change in the susceptible class is given now by the expression: I (t) dS(t) = Λ − λC(T (t))S(t) − μS(t), dt T (t)

(8.7)

with 

t

λC(T (x))S(x) 0

I (x) −μ(t−x) e PI (t − x)dx T (x)

representing the number of individuals who have been infected from time 0 to t and are still in class I . The discount factor exp(−μ(t − x)) takes into account removals due to deaths by natural causes (not HIV). Hence, if I0 (t) denotes individuals in class I at time t = 0 that are still infective at time t then the total number of infectives at time t is given by  t I (x) −μ(t−x) e PI (t − x)dx, (8.8) I (t) = I0 (t) + λC(T (x))S(x) T (x) 0 where I0 (t) is assumed (for biological and mathematical reasons) to have compact support (vanishing for large enough t). The expression for A(t) turns out to be the sum of three terms: A0 e−(μ+d)t , where A0 = A(0), representing individuals who had full-blown AIDS at time zero and are still alive; A0 (t) representing individuals initially in class I who moved into class A and are still alive at time t; and those who joined the I class after time t = 0 (see below). We assume that A0 (t) approaches zero as t approaches infinity. The term representing infected individuals “born” after time t = 0 is given by 1  τ % t I (x) −μ(τ −x) ˙ e λC(T (x))S(x) [−PI (τ − x)e−(μ+d)(t−τ ) ]dx dτ, T (x) 0 0 where −P˙I (τ − x), denotes the rate of removal from the class I at time τ or (τ − x) units after infection. Therefore 1  τ % t I (x) −μ(t−x) ˙ −(t−τ ) [−PA (τ − x)e ]dx dτ A(t) =p e λC(T (x))S(x) T (x) 0 0 + A0 e−(μ+d)t + A0 (t).

(8.9)

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8 Models for HIV/AIDS

The model given by equations (8.7) is a system of nonlinear integral equations. The standard results on well-posedness for these systems, as found in [82] guarantee the existence and uniqueness of solutions and their continuous dependence on parameters. The proof of positivity is given in [32]. The basic reproduction number R0 ) of the system (8.7) is given by R0 = λC



Λ μ



∞

PI (x)e−μx dx,

(8.10)

0

where 

∞

PI (x)e−μx dx,

0

is the death-adjusted mean infective period, D. If PI (x) = e−αI x then (8.10) reduces to (8.5). We also observe that as before DI =



∞

PI (s)e−μs ds

0

denotes the mean infective periods of the class I . System (8.7) with I0 (t) = 0 always has the equilibrium (S, I ) =



 Λ ,0 , μ

(8.11)

but no other constant solutions. However, since I0 (t) must be zero for large t, one would expect, under appropriate assumptions, that (Λ/μ, 0) will be an attractor or “asymptotic equilibrium” as t → +∞. The following results have been shown in [32, 35]. Theorem 8.3 The infection-free state (Λ/μ, 0) of the limiting system (8.7) is a global attractor; that is, limt→+∞ (S(t), I (t)) = (Λ/μ, 0) for any positive solution of system (8.7) as long as R0 ≤ 1. Theorem 8.4 The infection-free state of system (8.7) is unstable when R0 > 1 and there exists a constant W ∗ > 0, such that, any positive solution (S(t), I (t)) of (8.7) satisfies lim supt→+∞ I (t) ≥ W ∗ . In other words, if R0 > 1 the disease-free state (8.4) cannot be an attractor of any positive solution. That is, every solution has at least approximately W ∗ infectives (this W ∗ is the same as that in the statement of Theorem 8.5 below) for a sequence of times t tending to +∞ and if S(t), I (t) approach nonzero constants as t → +∞, when R0 > 1 then the results in [82] guarantee that these constants must satisfy

8.4 An Age of Infection Model

281

the limiting system associated with (8.7), which is given by the following set of equations: dS I (t) = Λ − λC(T (t))S(t) − μS(t) dt T (t)  t I (x) −μ(t−x) I (t) = PI (t − x)dx. λC(T (x))S(x) e T (x) 0

(8.12)

The limiting system (8.12) is an autonomous system for which we have established the following result: Theorem 8.5 If R0 > 1 the limiting system (8.12) has a unique positive equilibrium S ∗ , I ∗ . If in addition (d/dT )(C(T )/T ) ≤ 0, then this endemic equilibrium is locally asymptotic. Theorem 8.5 indicates that there is a switch of stability from (Λ/μ, 0) to (S ∗ , I ∗ ) as R0 crosses 1. We also conjecture but have not proved that the asymptotic dynamics of system (8.7) and the limiting system (8.12) agree. An alternative approach can be found in [61]. The proofs of these results can be found in [35].

8.4 An Age of Infection Model The model presented here is developed in [103]. We consider a homogeneouslymixing male homosexual population with infected members stratified by infection age (time since having been infected). We divide the population into three compartments: S (uninfected, but susceptible), I (HIV infected but with minimal or no symptoms), and A (fully developed AIDS). We assume members of the class A are no longer sexually active, and we let T = S + I be the size of the sexually active population. We let t denote time and τ denote age of infection, and we stratify the infected population by writing I (t) =



∞

i(t, τ )dτ,

0

where i(t, τ ) denotes the infection age density at time t. We assume: • the mean number of sexual contacts per individual in unit time is a, • there is a mean transmission rate λ(τ ) at which a typical susceptible individual contracts the infection by contact with an infected individual of infection age τ , • there is a rate α(τ ) of leaving the sexually active population (because of progression to AIDS) that depends on the age of infection, • there is a constant rate of recruitment Λ into the sexually active population,

282

8 Models for HIV/AIDS

• there is a constant rate μ of departure of uninfected members from the sexually active population, • there is a constant death rate ν from full-blown AIDS. Under these assumptions, the fraction of members remaining in the class I τ time units after having been infected is τ

P (τ ) = e−μτ −

0

α(σ )dσ

.

Then i(t, τ ) = i(t − τ, 0)P (τ ). We define the total infectivity at time t, W (t) = W0 (t) +



0

t

λ(τ )i(t, τ )dτ = W0 (t) +



t 0

λ(τ )i(t − τ, 0)P (τ )dτ,

where W0 (t) is the infectivity at time t of those individuals who were infected at time t = 0. Then the rate of new infections in unit time is B(t) = i(t, 0) = a

S(t) W (t), T (t)

and W (t) = W0 (t) +



t 0

λ(τ )P (τ )B(t − τ )dτ.

We will take a to be constant, but one could assume more generally that a is a function of the total sexually active population size T . These assumptions lead to the model S ′ (t) = Λ − B(t) − μS(t)  t W (t) = W0 (t) + λ(τ )P (τ )B(t − τ )dτ 0

S(t) W (t) T (t)  t I (t) = I0 (t) + B(t − τ )P (τ )dτ.

B(t) = a

0

(8.13)

8.4 An Age of Infection Model

283

Since we wish to study equilibria and their stability, we consider the limit system of (8.13), namely S ′ (t) = Λ − B(t) − μS(t)  ∞ W (t) = λ(τ )P (τ )B(t − τ )dτ

(8.14)

0

B(t) = a I (t) =



S(t) W (t) T (t) ∞ 0

B(t − τ )P (τ )dτ.

In order to obtain an expression for the number of active AIDS cases, not part of the model since individuals in the class A are assumed not to have any further sexual contacts, but included because it provides a relation that may be compared with data, we differentiate the equation  t I (t) = B(s)P (t − s)ds 0

of (8.14), using P ′ (u) = −[μ + α(u)]P (u). We obtain ′

I (t) = B(t) − μI (t) −



t

0

B(s)α(t − s)P (t − s)ds.

The input to the AIDS class A is  t B(s)α(t − s)P (t − s)ds. 0

Thus the number of active AIDS cases is given by  ∞ A′ (t) = α(t − s)P (t − s)B(s)ds − νA(t). 0

Analysis of the model (8.14) would be considerably simpler if we had assumed mass action incidence rather than standard incidence, because use of standard incidence brings T (t) = S(t)+I (t) into the model. However, mass action incidence is much less plausible for sexual transmission models than standard incidence. For the model it is not difficult to show that the basic reproduction number is given by R0 = a



0

∞

λ(τ )P (τ )dτ,

284

8 Models for HIV/AIDS

and that there is a disease-free equilibrium S = Λ/μ, I + B = W = 0 which is asymptotically stable if R0 < 1. Calculation of the endemic equilibrium is more difficult, but it is possible to show that there is an endemic equilibrium that is asymptotically stable at least for values of R0 larger than 1 but close to 1. For larger values of R0 the endemic equilibrium may be unstable, and there may be a Hopf bifurcation [64] and sustained oscillatory solutions of the model.

8.5 *HIV and Tuberculosis: Dynamics of Coinfections HIV diminishes the ability of the immune system to respond to invasions by infectious agents such as M. tuberculosis. Furthermore, as HIV infection progresses, immunity often declines with patients becoming more susceptible to typical or rare infections. In wealthier societies HIV and TB treatments are common; these drugs have altered significantly the joint dynamics of TB and HIV. The modeling literature on the independent dynamics of HIV or TB is quite extensive. TB efforts include, for example, [9, 18, 40, 42, 43, 50, 51, 89] while HIV/AIDS include [31, 63, 80, 103] to name a few more. TB/HIV coinfection modeling efforts have also been published. Kirschner [73] developed an immunological model describing HIV-1 and TB coinfections within a host. Naresh et al. [86] introduced a model involving a population sub-divided into four epidemiological classes: susceptible, TB infective, HIV infective, and those with AIDS; a model focusing on the transmission dynamics of HIV and treatable TB in variable size populations. Schulzer et al. [101] looked at HIV/TB joint dynamics using actuarial methods. West and Thompson [105] introduced models for the joint dynamics of HIV and TB that were explored via numerical simulations; their main goal was to estimate parameters and use their estimates to forecast the future transmission of TB in the United States. Porco et al. [90] looked, using a discrete event simulation model, at the impact of HIV on the probability and expected severity of TB outbreaks. Additional efforts include those in [91, 98]. A system of differential equations is used in [92] to model the joint dynamics of TB and HIV. The total population is divided into the following epidemiological subgroups: S, susceptible; L, latent with TB; I , infectious with TB; T , successfully treated with TB; J1 , HIV infectious; J2 , HIV infectious and TB latent; J3 , infectious with both TB and HIV; and A, “full-blown” AIDS. The compartmental diagram in Fig. 8.2 illustrates the flow of individuals as they face the possibility of acquiring specific-disease infections or even coinfections. The TB/HIV model is given by the following systems of eight ordinary differential equations:

8.5 *HIV and Tuberculosis: Dynamics of Coinfections

285

Fig. 8.2 Transition diagram between classes for the dynamics of TB and HIV coinfections. The force of infection for TB is λT = c(I + J3 )/N , and the force of infection for HIV is λH = σ J ∗ /R, where J ∗ = J1 + J2 + J3

I + J3 J∗ dS = Λ − cS − σS − μS dt N R

TB :

dL I + J3 J∗ = c(S + T ) − σL − (μ + k + r1 )L dt N R dI = kL − (μ + d + r2 )I dt

(8.15a)

dT J∗ I + J3 = r1 L + r2 I − cT − σT − μT , dt N R dJ1 J∗ I + J3 = σ (S + T ) − cJ1 − (α1 + μ)J1 + r ∗ J2 dt R N

HIV :

J∗ I + J3 dJ2 = σL + cJ1 − (α2 + μ + k ∗ + r ∗ )J2 dt R N dJ3 = k ∗ J2 − (α3 + μ + d ∗ )J3 dt dA = α1 J1 + α2 J2 + α3 J3 − (μ + f )A, dt

(8.15b)

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8 Models for HIV/AIDS

Table 8.1 Definition of parameters and state variables used in the TB/HIV model (8.15) Symbol N R J∗ Λ c σ μ k k∗ d d∗ f r1 r2 r∗ αi

Definition Total population Total active population (= N − I − J3 − A = S + L + T + J1 + J2 ) Individuals with HIV who have not developed AIDS (= J1 + J2 + J3 ) Constant recruitment rate Transmission rate of TB Transmission rate of HIV Per-capita natural death rate Per-capita TB progression rate for individuals not infected with HIV Per-capita TB progression rate for individuals infected also with HIV Per-capita TB-induced death rate Per-capita HIV-induced death rate Per-capita AIDS-induced death rate Per-capita latent TB treatment rate for individuals with no HIV Per-capita active TB treatment rate for individuals with no HIV Per-capita latent TB treatment rate for individuals with also HIV Per-capita AIDS progression rate for individuals in the Ji (i = 1, 2, 3)

where N = S + L + I + T + J1 + J2 + J3 + A, R = N − I − J3 − A = S + L + T + J1 + J2 , J ∗ = J1 + J2 + J3 .

(8.16)

The variable R here collects non-infectious “circulating” individuals, that is, those who do not have active TB or AIDS. Definitions of model parameters are collected in Table 8.1. The modeling assumptions include: homogenous mixing; HIV positive and TB infective (J3 ) showing severe HIV symptoms cannot be effectively treated for active TB; TB infections are only acquired through contacts with TB infectious individuals (I and J3 ); and individuals may acquire HIV infections only through contacts with HIV infectious individuals (J ∗ group). Further, the “probability” of infection per contact is assumed to be the same for T and S classes (β and λ). Furthermore, I (TB infectious), J3 (TB and HIV infectious), and A (AIDS) individuals are too ill to remain sexually active and, consequently, they do not transmit HIV through sexual activity. Hence, R ≡ N − I − J3 − A and the HIV incidence is modeled by σ J ∗ /R (see [29, 74, 108]). The probability of having a contact with HIV infectious individuals is modeled as J ∗ /R and the number of new HIV infections in a unit time is therefore σ SJ ∗ /R [IV drug injections, vertically-transmitted HIV (children of birth), or HIV transmission via breast feeding, forms of HIV transmission are ignored]. The most drastic in this model comes from the incorporation of sexual transmission as an indirect risk

8.5 *HIV and Tuberculosis: Dynamics of Coinfections

287

factor, a function of HIV prevalence. Further, demographic changes are ignored or alternatively, it is assumed that the time scale under consideration is such that changes in population size are not too significant. The TB control reproduction number (under treatment) is given by the expression R1 =

ck (μ + k + r1 )(μ + d + r2 )

(8.17)

while the HIV reproduction number is R2 =

σ . α1 + μ

(8.18)

R1 is the product of the average number of susceptible infected by one TB infective individual over its effective infective period, c/(μ+d+r2 ), and the fraction of the population that survives the TB latent period, k/(μ + k + r1 ). R1 denotes the number of secondary TB infectious cases generated by a typical TB infective individual during its effective infective period if introduced in a population of mostly TB-susceptible individuals, in a population where TB treatment is accessible. R2 is the HIV reproduction number in a TB-free society, the number of secondary HIV infections produced by an HIV infectious (but not TB-infected) individual during its infectious period if introduced in a population of HIV-susceptible individuals (in a TB-free world). The reproduction numbers do not involve the parameters tied in to the dynamics of TB-HIV coinfection, that is, k ∗ and α3 . Consequently, the reproduction number for system (8.15) under TB treatment is given by R = max{R1 , R2 }. We have shown in [92] that TB and HIV will die out if R < 1 while either or both diseases may become endemic if R > 1. In [92], it was shown that system (8.15) is well-posed, that is, solutions that start in this octant where all the variables are non-negative stay there. It was also shown that system (8.15) has three possible non-negative boundary equilibria: the diseasefree equilibrium (DFE) or E0 , the TB-only (HIV-free) equilibrium or ET , and the HIV-only (TB-free) equilibrium or EH . The components of E0 are S0 =

Λ , L0 = I0 = T0 = J01 = J02 = J03 = A0 = 0. μ

The ET components are Λ IT , LT = , μ + cIT /NT R1b = J2T = J3T = AT = 0,

ST = J1T

IT =

NT (R1 − 1) , R1 + R1a

TT =

(r1 L + r2 IT )ST , Λ

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8 Models for HIV/AIDS

where NT =

Λ , μ + d(R1 − 1)/(R1 + R1a )

with R1a =

c k , R1b = . μ + k + r1 μ + d + r2

(8.19)

The EH components are SH =

Λ , μR2 + α1 (R2 − 1)

J1H = (R2 − 1)SH ,

LH = IH = TH = 0,

J2H = J3H = 0,

AH =

α1 J1H . μ+f

The following results were established in [92]: Theorem 8.6 The disease-free equilibrium E0 is locally asymptotically stable if R < 1, and it is unstable if R > 1. Theorem 8.7 The HIV-free equilibrium ET is locally asymptotically stable if R1 > 1 and R2 < 1. We observe that EH may not be locally asymptotically stable under the conditions R1 < 1 and R2 > 1. Our numerical studies show that when R1 < 1 and R2 > 1 it is possible that the equilibrium EH is unstable and TB can coexist with HIV [92]. Further, whenever both reproduction numbers are greater than 1, that is, R1 > 1 and R2 > 1, ET and EH both exist and E0 is unstable. Our numerical studies show that all three boundary equilibria are unstable and solutions converge to an interior equilibrium point. Furthermore, partial analytical results and numerical simulations support the existence of an interior equilibrium Eˆ when both reproduction numbers, R1 and R2 , are greater than 1. The numerical simulations of the system further suggest that the interior equilibrium is LAS in most cases although the possibility of stable periodic solutions seems likely [92]. When both reproduction numbers are greater than 1, i.e., R1 > 1 and R2 > 1, ET and EH both exist and E0 is unstable. In this case, the numerical simulations of the model show that it is possible that all three boundary equilibria are unstable and solutions converge to an interior equilibrium point. Although explicit expressions for an interior equilibrium are very difficult to compute analytically, we have managed to obtain some relationships that can be used to determine the existence of an interior equilibrium.

8.5 *HIV and Tuberculosis: Dynamics of Coinfections

289

ˆ L, ˆ Iˆ, Jˆ1 , Jˆ2 , Jˆ3 , A) ˆ denote an interior equilibrium with all compoLet Eˆ = (S, nents positive, and let x and y denote the fractions: x=

Iˆ + Jˆ3 >0 Nˆ

and

y=

Jˆ∗ > 0. Rˆ

(8.20)

Note that x and y correspond to the levels of disease prevalence for TB and HIV, respectively. By setting the right-hand-side of the system (8.15) equal to zero we can obtain the following two equations for x and y: x = xF (x, y), y = yG(x, y),

(8.21)

where  ˆ k Sˆ k ∗  Sσy c + + Jˆ1 , Δ2 Δ3 B1 Nˆ (μ + d + r2 )B1 σ # 1 ˆ r ∗ Lˆ  cx  k ∗  Lˆ  k ∗ $ G(x, y) = S + Tˆ + 1+ 1+ + 1+ , Δ2 Δ2 Δ3 Δ2 Δ3 Rˆ B2 (8.22) F (x, y) =

in which Sˆ = Tˆ =

Λ , μ + cx + σy r1 +

r2 k μ+d+r2

cx + σy + μ

cΛ x, B1 (μ + cx + σy)

∗ˆ Sˆ + Tˆ + rΔL2 σy ˆ , J1 = B2

Lˆ =

ˆ L,

ˆ + Jˆ1 cx) k ∗ (Lσy Jˆ3 = , Δ2 Δ3

Aˆ =

Iˆ =

k ˆ L, μ + d + r2

ˆ Lσy + Jˆ1 cx Jˆ2 = , Δ2

(8.23)

 1 ˆ α1 J1 + α2 Jˆ2 + α3 Jˆ3 , μ+f

and Δ2 = α2 + μ + k ∗ + r ∗ , Δ3 = α3 + μ + d, B1 = σy + μ + k + r1 −

cx(r1 +

r2 k μ+d+r2 )

cx + σy + μ ≥ σy + μ + k + r1 − (r1 + k) > 0, cx(α1 + μ + k ∗ ) + α1 + μ. B2 = Δ2

(8.24)

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8 Models for HIV/AIDS

Fig. 8.3 Contour plots showing the intersection points of the curves F (x, y) = 1 (dashed curve) and G(x, y) = 1 (solid curve) for various values of R2 with R1 being fixed at 1.5 (c = 12). The values of R2 in (A)–(C) are 3.6, 4.6, and 7, respectively (corresponding to λσ = 0.41, 0.52, and 0.8). The axes are x = (I + J3 )/N and y = J ∗ /R, representing the factors in the ˆ

ˆ

incidence functions for TB and HIV, respectively. The intersection (x, ˆ y) ˆ = ( I +ˆJ3 , N components of the interior equilibrium Eˆ if 0 < xˆ < 1 and yˆ > 0

Jˆ∗ ) determines Rˆ

Note that x > 0 and y > 0, Eq. (8.21) reduces to F (x, y) = 1,

G(x, y) = 1,

(8.25)

and an intersection of the two curves determined by Eq. (8.25), denoted by xˆ and y, ˆ corresponds to a coexistence equilibrium of TB and HIV. We can consider xˆ as a measure for the TB prevalence. The intersection property of the two curves given by F (x, y) = 1 and G(x, y) = 1 are illustrated in Fig. 8.3. ˆ y) ˆ of the contour plots of F (x, y) = 1 Figure 8.3 plots the intersection point (x, (dashed curve) and G(x, y) = 1 (solid curve) for several values of R2 with R1 being fixed (R1 = 1.5 corresponding to c = 12). Again, an interior equilibrium Eˆ can be determined by xˆ and yˆ if 0 < xˆ < 1 and yˆ > 0. This figure illustrates how xˆ changes with increasing R2 . We have chosen k ∗ = 5k (i.e., the progression rate to active TB in individuals with both latent TB and HIV is five times higher than that in individuals with latent TB only), α3 = 5α1 (i.e., the progression to AIDS in individuals with active TB is five times higher than that in individuals without TB). For this set of parameter values, the values of R2 in Fig. 8.3A–C are 3.6, 4.6, and 7, respectively. It shows that when R2 increases from 3.8 to 4.6, the F (x, y) = 1 curve does not change much while the right-end of the G(x, y) = 1 curve moves to the right of the F = 1 curve. This leads to an intersection point of the two curves (see (A) and (B)), which corresponds to an ˆ When R2 is further increased to 7, the G(x, y) = 1 curve interior equilibrium E. changes from decreasing to increasing (see (C)). Although there is still a unique intersection point, the y = Jˆ∗ /Rˆ component may become greater than 1. This is still biologically feasible as J /R can exceed 1 (see (C)). The intersection points in (A)ˆ ˆ ˆ∗ (C) are (x, ˆ y) ˆ = ( I +ˆJ3 , Jˆ ) = (0.15, 0.07), (0.25, 0.4), (0.33, 1.25), respectively. N R We observe that xˆ increases with R2 from 0.15 to 0.33. This implies that the prevalence of HIV may have significant impact on the infection level of TB.

8.5 *HIV and Tuberculosis: Dynamics of Coinfections

291

Fig. 8.4 Time plots of prevalence of TB and HIV. The TB curves (solid) represent the fraction of active TB ((I + J3 )/N ), and the HIV curve (dashed) represents the activity-adjusted fraction of HIV (J ∗ /R)

Figure 8.4 examines changes in infection levels over time. It plots the time series of [I (t) + J3 (t)]/N (t) (fraction of active TB) and J ∗ (t)/R(t) (activity-adjusted fraction of HIV infectious) for fixed R1 and various R2 . The top two figures are for the case when the reproduction number for TB is less than 1 (R1 = 0.96 < 1 or c = 7.5), and the reproduction number for HIV is R2 = 0.9 < 1 (or σ = 0.105) in (a) and R2 = 1.3 > 1 (or σ = 0.15) in (a). It illustrates in Fig. 8.4a that TB cannot persist if R2 < 1. However, if R2 > 1 then it is possible that TB can become prevalent even if R1 < 1 (see Fig. 8.4b). The bottom two figures are for the case when the reproduction number of TB is greater than 1 (R1 = 1.2, or c = 9.1), and R2 = 2 (or σ = 0.23) in (c) and R2 = 3 (or σ = 0.34) in (d). It demonstrates that an increase in R2 will lead to an increase in the level of TB prevalence as well. All other parameters are the same as in Fig. 8.3 except that k ∗ = 3k. Another way to look at the role of HIV on TB dynamics is to compare the outcomes between the cases where HIV is absent or present (instead of varying the value of R2 ). This result is presented in Fig. 8.5. The reproduction numbers are identical in Fig. 8.5A, B: R1 = 0.98 < 1 (c = 7.7) and R2 = 1.2 > 1 (σ = 0.137). Other parameter values are the same as in Fig. 8.4 except that k ∗ = k. The variables plotted are (I + J2 )/N and J ∗ /N. Figure 8.5A is for the case when HIV is absent by letting J ∗ (0) = 0. It shows that TB cannot persist. In Fig. 8.5B, the initial value of HIV is positive (i.e., J ∗ (0) > 0). It shows that both TB and HIV coexist. Examples of other mathematical models on dynamics of TB/HIV coinfections include [73, 86, 90, 91, 101].

292

8 Models for HIV/AIDS

Fig. 8.5 For the plot in (A), HIV is absent by letting J ∗ (0) = 0. It shows that TB cannot persist. In (B), the initial value of HIV is positive (i.e., J ∗ (0) > 0). It shows that both TB and HIV will coexist

8.6 *Modeling the Synergy Between HIV and HSV-2 The example presented in this section considers the synergy between HIV and HSV-2. One of the questions that is of interest for public health officials is how treatment of HSV-2 may influence the prevalence and control of HIV. Several mathematical models have been developed to investigate the transmission dynamics of HSV-2 (e.g., [17, 53, 87, 100] and references therein) and HIV (e.g., [12–14, 44, 84, 85] and references therein). To our knowledge, however, there have only been a few modeling studies of the epidemiological synergy between HSV-2 and HIV. Using the individual-based model STDSIM, White et al. [106] studied the population-level effect of HSV-2 therapy on the incidence of HIV in sub-Saharan Africa. Foss et al. [54] developed a dynamic HSV/HIV model to estimate the contribution of HSV-2 to HIV transmission from clients to female sex workers in southern India and the maximum potential impact of “perfect” HSV-2 suppressive therapy on HIV incidence. Blower and Ma [16] used a transmission model that specifies the dynamics of HIV and HSV-2 to predict the effect of a highprevalence HSV-2 epidemic on HIV incidence. Abu-Raddad et al. [1] constructed a deterministic compartmental model to describe HIV and HSV-2 transmission dynamics and their interaction. However, the model studied in [16] does not include heterogeneity in sexual activity and assumes that individuals mix randomly, whereupon each infective individual is equally likely to spread the disease to all others. Also, gender is not incorporated into the models studied by either [16] or [1]. The models in [54, 106] incorporate various heterogeneities, including gender and/or age, but not sexual activity, and only numerical simulations are conducted. Gender may be an important factor in modeling the epidemiological synergy between HSV-2 and HIV as shown in the meta-analysis of several studies that male parameters differ from the corresponding female parameters. For example, the male-to-female HSV-2 transmission probability is greater than the female-to-male transmission probability [45, 104], and thus the risk of female-to-male transmission per sex act is less than the risk of male-to-female transmission [84, 85]. Thus, to fully understand the epidemiological synergy between HSV-2 and HIV and to investigate

8.6 *Modeling the Synergy Between HIV and HSV-2

293

measures for controlling these sexually transmitted diseases, it is important to analyze models that consider heterogeneities in sexual activity, mixing within and between different activity groups and genders. In [2, 52], a model incorporating both HIV and HSV-2 infections was analyzed. The model considers one male population and multiple female populations based on their activity levels with variable male preferences to different female groups. Results from the model demonstrate that the heterogeneity in activity levels and male preference in mixing may play an important role in model outcomes. More details of the model analysis are presented below. Consider a population consisting of sexually active female and male individuals. Consider the case in which the female population is divided into subgroups based on levels of sexual activity (e.g., number of partners) with a low-risk group (e.g., members of the general population) and a high-risk group (e.g., sex workers), while all individuals in the male population have the same activity level. These subpopulations are labeled by the subscripts f1 , f2 , m, which denote low- and high-risk females and males, respectively. Let Ni denote the population sizes of groups i, where i = m, f1 , f2 . The population in each group is assumed to be homogeneous in the sense that individuals have the same infectious period, duration of immunity, contact rate, and so on. We divide the progression of HIV into two stages, acute infection and AIDS. Similarly, HSV-2 is represented by acute and latent infection stages. Because individuals infected with HIV alone or HSV-2 alone can become coinfected with both HIV and HSV-2, each group i (i = m, f1 , f2 ) is further divided into seven epidemiological classes or subgroups: susceptible, infected with acute HSV-2 only (Ai ), infected with latent HSV-2 only (Li ), infected with HIV only (Hi ), infected with HIV and acute HSV-2 (Pi ), infected with HIV and latent HSV-2 (Qi ) and AIDS (Di ). A transition diagram between these epidemiological classes within group i is depicted in Fig. 8.6.

Fig. 8.6 Transition diagram of the coupled dynamics between HIV and HSV-2. The top row includes classes infected with HSV-2 only, and the bottom row includes classes infected with either HIV only or coinfected with HIV and HSV-2

294

8 Models for HIV/AIDS

For each sub-population i (i = f1 , f2 , m) there is a per-capita recruitment rate μi into the susceptible group. For all classes there is a constant per-capita rate μi of exiting the sexually active population. Thus, the total population Ni in group i remains constant for all time. Susceptible people in group i acquire infection with H HSV-2 or HIV at the rate λA i (t) or λi (t), respectively. Upon being infected with HSV-2, people in group i enter the class Ai (infected with acute HSV-2 only). These individuals become latent Li at the constant rate ωiA (an average duration in Ai is 1/ωiA ). Following an appropriate stimulus in individuals with latent HSV2, reactivation may occur [17]. We assume that people with latent HSV-2 only reactivate at the rate γiL . Individuals with HIV are assumed to develop AIDS at the rate diH . Let δiA and δiL denote the enhanced susceptibility to HIV infection for individuals in group i with acute or latent HSV-2 infection. Classes Pi and Qi are similar to Ai and Li , respectively, except that Ai and Li denote individuals with HSV-2 only whereas Pi and Qi denote individuals with coinfections. The difference in stage durations is indicated by the superscripts (e.g., 1/γiL for the Q L class and 1/γi for the Q class). Finally, the antiviral treatment rates for the Ai and Pi individuals are denoted by θiA and θiQ , respectively. Because antiviral medications will also suppress reactivation of latent HSV-2, we assume that the Q reactivation rate of people with latent HSV-2 γiL (or γi ) is a decreasing function of Q θiA (or θiP ), denoted by γiL (θiA ) (or γi (θiP )). The sources for most of the parameter values are from [1, 53] (see [52] for more details). Based on Fig. 8.6, Alvey et al. [2] studied the following model: dSi dt dAi dt dLi dt dHi dt dPi dt dQi dt

H = μi Ni − (λA i (t) + λi (t))Si − μi Si , L A A H A A = λA i (t)Si + γi (θi )Li − δi λi (t)Ai − (ωi + θi + μi )Ai , L A = (ωiA + θiA )Ai − δiL λH i (t)Li − (γi (θi ) + μi )Li , H A H = λH i (t)Si − δi λi (t)Hi − (μi + di )Hi , Q

H A P P P P = δiA λH i (t)Ai + δi λi (t)Hi + γi (θi )Qi − (ωi + θi + μi + di )Pi , Q

Q

P P P = δiL λH i (t)Li + (ωi + θi )Pi − (γi (θi ) + μi + di )Qi , i = m, f1 , f2 , (8.26) j

where the functions λi (t) represent the forces of infection given below. Let bi (i = m, f1 , f2 ) be the rate at which individuals in group i acquire new sexual partners (also referred to as contact rates), and let cj denote the probability that a male chooses a female partner in group j (j = f1 , f2 ). Then c1 + c2 = 1. For ease of notation, let c1 = c,

c2 = 1 − c.

8.6 *Modeling the Synergy Between HIV and HSV-2

295

Overall, the number of female partners in groups j (j = f1 , f2 ) that males acquire should be equal to the number of male partners that females in groups j acquire. These observations lead to the following balance conditions: bm cNm = bf1 Nf1 ,

bm (1 − c)Nm = bf2 Nf2 .

(8.27)

To ensure that constraints in (8.27) are satisfied, we assume in numerical simulations that bm and c are fixed constants with bf1 and bf2 being varied according to Nm , Nf1 , and Nf2 . The force of infection functions can be expressed as λH m (t) =

2

Q

bm ci βfHi m

Nfi

i=1

2

,

Q

H λH fj (t) = bfj βmfj

λA m (t) =

Hfi + δfPi Pfi + δfi Qfi

PP + δ Q Hm + δm m m m , Nm

bm ci βfAi m

i=1

A λA fj (t) = bfj βmfj

Afi + σfPi Pfi Nfi

Am + σmP Pm , Nm

j = 1, 2, (8.28)

,

j = 1, 2,

where Ni = Si + Ai + Li + Hi + Pi + Qi , i = m, f1 , f2 H (β H ), i = f , f are denotes the total population size of group i. In (8.28), βim 1 2 mi the HIV transmission probabilities per partner between females infected with HIV in group i and susceptible males (between males infected with HIV and susceptible A (β A ), i = f , f are the HSV-2 transmission probabilities females in group i); βim 1 2 mi per partner between females infected with acute HSV-2 in group i and susceptible males (between males infected with acute HSV-2 and susceptible females in group Q i); δiP and δi (i = m, f1 , f2 ) are the enhanced HIV infectiousness of coinfected individuals, and σiP (i = m, f1 , f2 ) are the enhanced HSV-2 infectiousness of coinfected individuals.

8.6.1 Reproduction Numbers for Individual Diseases For each of the two diseases, we can compute the reproduction number in the absence of the other disease. Let R0A and R0H denote these reproduction numbers for HSV-2 and HIV, respectively. Due to the loop between the symptomatic and

296

8 Models for HIV/AIDS

asymptomatic stages of HSV-2, the derivation of analytical expression for R0A for model (8.26) is not straightforward. A detailed derivation of the following formula for R0A can be found in [2, 52]: R0A =

0

A Rmf 1m

2

2  A , + Rmf m 2

(8.29)

where A Rmf jm

. . =/

A bfj βmf j A + θA + μ ωm m m

· PmA ·

bm cj βfAj m ωfAj + θfAj + μfj

· PfAj ,

j = 1, 2

with PiA (i = m, f1 , f2 ) representing the probability that an individual of group i is in the acute stage (A), which is given by PiA





A ωi + θiA + μi γiL (θiA ) + μi  , =  L A γi (θi ) + ωiA + θiA + μi μi

i = m, f1 , f2 .

(8.30)

The formulas for PiA in (8.30) can be explained as follows. Let p=

ωiA + θiA

ωiA

+ θiA

+ μi

,

q=

γiL γiL

+ μi

,

where p represents the probability that an individual moves from the acute stage (A) to the latent stage (L), and q represents the probability that an individual moves from L to A. Thus, the probability that an individual is in the acute stage within the A ⇋ L loop is 



A ωi + θiA + μi γiL + μi  = PiA . (pq) = L A + θA + μ μ γ + ω i i i i i k=1

∞

k

Notice that in the formula for R0A the balance conditions in (8.27) have been used. A Other factors in Rmf (i = 1, 2) also have clear biological interpretations: im A is the number of new infections that a male will cause in females of • bfj βmf j group j (j = 1, 2) per unit of time; • bm cj βfAj m is the number of new infections that a female in group j (j = 1, 2) will cause in males per unit of time; (i = m, f1 , f2 ) represents the mean time that an individual in group i • A 1A ωi +θi +μi

remains infected (i.e., in either A or L).

8.6 *Modeling the Synergy Between HIV and HSV-2

297

 A represents the average secondary HSV-2 male infections by one Thus, Rmf jm male individual through females in group j (j = 1, 2) while in the infectious stage (A) in a completely susceptible population. The square root is associated with the fact that we need to consider both the male-to-female and female-to-male processes to obtain the number of secondary infections. The overall reproduction number R0A A is an average of Rmf (i = 1, 2). im Let R0H denote the basic reproduction number for HIV in the absence of HSV-2. Then 0    H Rmf 1m

R0H =

2

2

H + Rmf 2m

,

where H Rmf jm

. H bm cj βfHj m . bfj βmf j =/ H · H , dm + μm df + μfj j

j = 1, 2.

H H The biological meanings of Rmf and Rmf can be explained in the similar way 1m 2m A A as those of Rmf1 m and Rmf2 m . It is clear that R0H represents the average secondary HIV male infections by one male individual (through both female groups) during the whole HIV infective period in a completely susceptible population.

8.6.2 Invasion Reproduction Numbers Let RAH denote the invasion reproduction number for HIV in a population where the HSV-2 infection is already established at the endemic equilibrium, which is denoted by E∂A . The nonzero components of E∂A are Si0 , A0i , and L0i , representing the density of susceptible, acute HSV-2, and HSV-2 latent, respectively, in group i. Let Ni0 = Si0 + A0i + L0i . For ease of notation, let λA0 m

= bm

2

ci βfAj m

i=1

A0fj Nf0j

,

A λA0 fj = bfj βmfj

A0m , Nm0

j = 1, 2

and   Q di = 1, δiP , δi ,

T  x0i = Si0 , δiA A0i , δiL L0i ,

i = m, f1 , f2 .

Note that the system (8.26) has 9 infected variables with HIV (Hi , Pi , Qi , i = m, f1 , f2 ). Consider the HIV-free equilibrium E∂A of system (8.26). The matrices

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8 Models for HIV/AIDS

F H and V H (corresponding to the new infection and remaining transfer terms, respectively) are given by ⎛

0

⎜ ⎜ H F H = ⎜ Fmf 1 ⎝ H Fmf2

FfH1 m FfH2 m

⎞

⎟ ⎟ 0 ⎟, ⎠ 0

0 0

⎛

VmH 0

0

⎞

⎜ ⎟ ⎜ ⎟ V H = ⎜ 0 VfH1 0 ⎟ , ⎝ ⎠ 0 0 VfH2

(8.31)

where FfHj m = bm cj βfHj m

x0m df , Nm0 j

H H Fmf = bfj βmf j j

x0fj Nf0j

dm ,

j = 1, 2

and ⎛

⎜ ⎜ ViH = ⎜ ⎝

(μi + diH + δiH λA0 i )

0

0

−δiH λA0 i

ωiP + θiP + μi + diP

−γi (θiP )

0

−(ωiP + θiP )

γiQ (θiP ) + μi + diQ

Q

⎞

⎟ ⎟ ⎟, ⎠

(8.32)

for i = m, f1 , f2 . Then, the next generation matrix for HIV, denoted by KH , can be expressed by KH = F H (V H )−1 ⎛ ⎞ 0 FfH1 m (VfH1 )−1 FfH2 m (VfH2 )−1 ⎜ ⎟

 ⎜ H ⎟ H )−1 (V 0 0 = ⎜Fmf ⎟ := kij 9×9 , m 1 ⎝ ⎠ H (V H )−1 Fmf 0 0 m 2

(8.33)

where the entries kij of the matrix KH can be found in the Appendix A of [52]. Noting that Rank(KH ) = 2 and that the sum of the diagonal elements in matrix KH is zero, it follows from Vieta’s formulas that if the numbers of susceptible people and those with acute and latent HSV-2 in group i are Si0 , A0i , L0i , respectively, the reproduction number for HIV infection is given by 0 0 H 0 RAH = R -A (Si , Ai , Li , 0, 0, 0) := ρ(KH ) = . 9 3 . . =/ kij kj i , i=1 j =4



−E2 (KH ) (8.34)

8.7 An HIV Model with Vaccination

299

Fig. 8.7 Numerical solutions of the system (8.26) for the case when R0A > 1, R0H < 1, and RAH > 1. The dashed and solid curves represent levels of HIV infections with and without HSV-2 present, respectively. It shows that even when the basic reproduction number for HIV R0H is less than 1, HIV can still persist if the invasion reproduction number RAH is greater than 1

where ρ(KH ) represents the spectral radius of the matrix KH and E2 (KH ) is the sum of all the 2 × 2 principal minors of matrix KH . It is shown in [52] that invasion is possible if and only if RAH > 1. A for HSV-2 to invade a population Similarly, an invasion reproduction number RH in which HIV is present (see [52]). Detailed results on the existence and local stability of the boundary equilibria can also be found in [52].

8.6.3 Influence of HSV-2 on the Dynamics of HIV Figure 8.7 illustrates the result of numerical simulations showing how the joint disease dynamics of HIV and HSV-2 may depend on the basic and invasion reproduction numbers. It is for the case when enhancement of HIV by HSV-2 is relatively strong with R0A > 1, R0H < 1, and RAH > 1. It shows that while HIV can invade and persist in the presence of HSV-2 (the dashed curve), it dies out in the absence of HSV-2 (the solid curve), suggesting that HSV-2 infection can favor the invasion of HIV.

8.7 An HIV Model with Vaccination Blower et al. [15] studied model for HIV with live attenuated HIV vaccines (LAHVs). Consider two viral strains, one wild strain and one vaccine strain. Divide the total population into the following epidemiological classes: susceptible individuals (S), unvaccinated individuals infected with the wild-type HIV (Iw ) or the vaccine strain (either by vaccination or by transmission) (Iv ) or dually infected

300

8 Models for HIV/AIDS

with both strains (Ivw ), and individuals with AIDS (A). The model consists of the following ordinary differential equations: S ′ = (1 − p)π − (cλv + cλW + μ)S,

Iv′ = pπ + cλv S − (1 − ψ)cλw Iv − (νv + μ)Iv , Iw′ = cλw S − (νw + μ)Iw ,

′ Ivw ′

(8.35)

= (1 − ψ)cλw Iv − (νvw + μ)Ivw ,

A = νw Iw + νv Iv + νvw Ivw − (μA + μ)A,

where λv and λw are per-capita risks of infection with the vaccine and wild-type strains, respectively, given by λv = βv

Iv , NSA

λw = βw

Iw + gIvw , NSA

and NSA = X + Iv + Iw + Ivw denotes the number of sexually active population. Other parameters include: βv and βw are infection rates for vaccine and wildtype strains, respectively, p is the fraction of new susceptibles vaccinated, π is the number of new susceptibles that join the sexually active population per unit time, c is the average rate of acquiring new sex partners, 1/μ is the average period of acquisition of new sex partners, 1/μA is the average survival time with AIDS, ψ denotes the degree of protection that the vaccine provides against infection with the wild-type strain, ν is the progression rate to AIDS in individuals infected with the LAHV strain (νv ), the wild-type strain (νw ), or both strains (νvw ), 1/μA is the average survival time from AIDS to death. The disease progression rates are related by the expression νvw = δνw , where δ specifies the vaccine-induced degree of reduction in the wild-type disease progression rate. A time-dependent uncertainty analysis of model (8.35) can be used [15] to predict the potential impact of LAHVs on the annual AIDS death rate, as illustrated in Fig. 8.8. It shows the result of infection with the wild-type strain of HIV for Zimbabwe (A) and Thailand (B), and the result of the LAHV strain for Zimbabwe (C) and Thailand (D). Parameter values used include the following probability density functions (pdfs): 1/μA (pdf: 9 months to 1 year to 18 months), βw (pdf: 0.05 to 0.1 to 0.2), βv = αβw where α (range of pdf: (0.001, 0.1)), νw (pdf: range from 50% progression to AIDS in 7.5 years to 50% progression in 10 years). Consider a mass vaccination campaign (with follow-up programs) that would vaccinate anywhere from 80% to 95% of susceptibles with p (range of pdf: (0.8, 0.95)), ψ (range of pdf: (0.5, 0.95)), δ (range of pdf: (0.1, 1)). The population size of sexually active adults are chosen to be 5,560,000 (Zimbabwe), 34,433,00 (Thailand).

8.8 A Model with Antiretroviral Therapy (ART)

301

Fig. 8.8 Simulation results of model (8.35). It plots annual AIDS deaths (per 100,000 individuals) for Zimbabwe (A), Thailand (B), Zimbabwe (C), and Thailand (D). Source: [15]

8.8 A Model with Antiretroviral Therapy (ART) A mathematical model with antiretroviral therapy (ART) is considered in [25] to study the effect of ART on risk behaviors and sexually transmitted infections (STI). Individuals are divided into two risk groups i = 1, 2, with s = 1 for STI+ and s = 0 for STI-. The population size for fix i and s is divided into the following epidemiological classes: susceptible to HIV (Sis ), untreated HIV+ (Iisu ), untreated with AIDS (Auis ), treated HIV+ (Iisτ ), treated with AIDS (Aτis ). The group sizes are Nis = Sis + Iisu + Iisτ + Aτis ,

i = 1, 2, s = 0, 1.

The per-capita rates of STI and HIV infection of a susceptible individual in risk group i are denoted by ξi (t) and λis (t), respectively, and are given by ξi (t) = θi

λi0 (t) = βi

2 j =1

j

Nj 1 , ρij s Nj s

i = 1, 2,



u τ τ s Ij s + (1 − η)(Ij s + Aj s )

, ρij s Nj s

λi1 = 3λi0 ,

(8.36)

i = 1, 2, (8.37)

302

8 Models for HIV/AIDS

where θi and βi are transmission rates of STI and HIV, respectively, for susceptibles of risk level i, ρij represents the sexual mixing between types i and j individual (e.g., proportionate mixing), η1 represents the reduction in HIV infectiousness due to treatment with ART. The following model is a simplified version of the model considered in [25]:     dSis = Λi − (λis (t) + μ)Sis + (1 − s) − ξi (t)Si0 + δSi1 + s ξi (t)Si0 − δSi1 , dt dIisu = λis (t)Sis − (γ u + μ + r h )Iisu dt     u u u u +ωIisτ + (1 − s) − ξi (t)Ii0 + δIi1 + s ξi (t)Ii0 − δIi1 dIisτ = r h Iisu − (γ τ + μ + ω)Iisτ dt     τ τ τ τ + s ξi (t)Ii0 − δIi1 +(1 − s) − ξi (t)Ii0 + δIi1

(8.38)

dAuis = γ u Iisu − (α u + μ + r a )Auis + ωAτis + (1 − s)δAui1 − sδAui1 dt dAτis = γ τ Iisτ − (α τ + μ + ω)Aτis dt     +r a Auis + (1 − s) − ξi (t)Aτi0 + δAτi1 + s ξi (t)Aτi0 − δAτi1 ,

where i = 1, 2, s = 0, 1, Λi is the recruitment rate to group i (i = 1, 2), γ u and γ τ are the rates of progression to AIDS for untreated and treated individuals, respectively, α u and α τ are the rates of AIDS mortality for untreated and treated individuals, respectively, δ is the recovery rate from the STI infection, η represents the reduction in HIV infectiousness as a result of ART, w is the withdraw rate from treatment, r a and r h are treatment coverage rates of AIDS and HIV-positive individuals, respectively, 1/μ represents the average duration of sexually active life. A more general model is studied in [25], in which a detailed model analysis is presented to demonstrated the impact of the wide-scale use of ART on HIV transmission.

8.9 Project: What If Not All Infectives Progress to AIDS? In the model (8.1) it is assumed that all HIV-infected individuals eventually progress to full-blown AIDS, as this appears to be the case. Suppose, however, that only a fraction p, 0 < p < 1 progress to AIDS while the remaining infectives remain in this class until they are no longer sexually active. In addition to the classes S, I , and A, a model must now also include a class Y of infective individuals that will not develop full-blown AIDS and a class Z of former Y -individuals who are no longer sexually active. The corresponding model is

8.9 Project: What If Not All Infectives Progress to AIDS?

303

Fig. 8.9 Flow diagram; single group model for the case when only a fraction of infected people will progress to AIDS

S(t)W (t) dS(t) = Λ − λC(T (t)) − μS(t) dt T (t) dI (t) S(t)W (t) = λpC(T (t)) − (αI + μ)I (t) dt T (t) S(t)W (t) dY (t) = λ(1 − p)C(T (t)) − (αY + μ)Y (t) dt T (t)

(8.39)

dA(t) = αI I (t) − (d + μ)A(t) dt dZ(t) = αY Y (t) − μZ(t) dt where W =I +Y

and

T = W + S.

(8.40)

A flow diagram is shown in Fig. 8.9. It is further assumed that individuals who develop full-blown AIDS are no longer actively infective, that is, that they have no sexual contacts; it is also assumed that infected individuals become immediately infective. Finally, it is assumed that individuals in this population become sexually inactive or acquire AIDS at the constant rates αY and αI (respectively) per unit time. Therefore, 1/(μ + αI ) gives the average or mean incubation period with the fraction 1/(μ + αY ) denoting the average or mean sexual life expectancy. For simplicity, we assume αI = αY , but it is possible to extend the model to the case αi = αY .

304

8 Models for HIV/AIDS

As before, the function C(T ) models the mean number of sexual partners an average individual has per unit time when the population density is T ; λ (a constant) denotes the average sexual risk per infected partner; λ is often thought as the product iφ [68], where φ is the average number of contacts per sexual partner and i the conditional probability of infection from a sexual contact when the latter is infected. Kingsley et al. [72] had presented (not surprising) evidence that the probability of seroconversion (infection) increases with the number of infected sexual partners. Hence, λC(T ) models the transmission rate per unit time per infected partner when the size of the sexually active population is T . We continue to assume C(T ) > 0,

C ′ (T ) ≥ 0,

(8.41)

Question 4 Show that for the model (8.39) the basic reproduction number is given by R0 = λ



   p Λ 1−p + , C σI σY μ

(8.42)

where σI = αI + μ, σY = αY + μ. R0 is given by the product of the three factors (epidemiological parameters): λ (the probability of transmission per partner), C(Λ/μ) (the mean number of sexual partners that an average susceptible individual has per unit time when everybody in the population is susceptible), and D=



 p 1−p . + σI σY

(8.43)

The death-adjusted mean infective period is D = pDI + (1 − p)DY with DI and DY denoting the death-adjusted mean infective periods, 1/σI and 1/σY of the I and Y classes, respectively. Question 5 Show that if R0 < 1, the disease-free equilibrium (Λ/μ, 0, 0) of the system (8.39) is asymptotically stable, and if R0 > 1 there is a unique endemic equilibrium (S ∗ , I ∗ , Y ∗ ), which is locally asymptotically stable, and the diseasefree state (Λ/μ, 0, 0) is unstable. Next, we allow arbitrary incubation period distributions by introducing two functions PI (s) and PY (s) representing the proportion of individuals who become I - or Y -infective at time t and that, if alive, are still infective at time t + s (survive as infectious). PI and PY , survivorship functions, are non-negative and non-increasing, and PI (0) = PY (0) = 1. It is further assumed that 

0

∞

PI (s)ds < ∞,



0

∞

PY (s)ds < ∞,

References

305

and thus, −P˙ (x) and −P˙Y (x) are the rates of removal of individuals from classes I and Y into classes A and Z, x time units after infection. Question 6 Derive the corresponding model and determine its basic reproduction number.

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57. Gupta S., R.M. Anderson, and R.M. May (1989) Networks of sexual contacts: implications for the pattern of spread of HIV, AIDS 3: 1–11. 58. Francis, D.P., P.M. Feorino, J.R. Broderson, H.M. Mcclure, J.P. Getchell, C.R. Mcgrath, B. Swenson, J.S. Mcdougal, E.L. Palmer, and A.K. Harrison (1984) Infection of chimpanzees with lymphadenopathy-associated virus, Lancet 2: 1276–1277. 59. Hadeler, K.P. (1989) Modeling AIDS in structured populations, 47th Session of the International Statistical Institute, Paris, August/September. Conf. Proc., C1-2: 83–99. 60. Hadeler, K.P. and C. Castillo-Chavez (1995) A core group model for disease transmission, Math. Biosc. 128: 41–55. 61. Hethcote, H.W., H.W. Stech, P. van den Driessche (1981) Nonlinear oscillations in epidemic models, SIAM J. Appl. Math. 40: 1–9. 62. Hethcote, H.W., J.W. van Ark (1987) Epidemiological methods for heterogeneous populations: proportional mixing, parameter estimation, and immunization programs. Math. Biosc. 84: 85–118. 63. Hethcote, H.W., and J.W. Van Ark (1992) Modeling HIV Transmission and AIDS in the United States, Lecture Notes in Biomathematics 95, Springer-Verlag, Berlin-Heidelberg-New York. 64. Hopf, E. (1942) Abzweigung einer periodischen Lösungen von einer stationaren Lösung eines Differentialsystems,Berlin Math-Phys. Sachsiche Akademie der Wissenschaften, Leipzig, 94: 1–22. 65. Hsu Schmitz, S.F. (1993) Some theories, estimation methods and applications of marriage functions and two-sex mixing functions in demography and epidemiology. Unpublished doctoral dissertation, Cornell University, Ithaca, New York, U.S.A. 66. Hsu Schmitz S.F. and C. Castillo-Chavez (1994) Parameter estimation. Brit. Med. J. 293: 1459–1462. 67. Huang, W., K.L.Cooke, and C. Castillo-Chavez, (1992) Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math. |textbf52: 835–854. 68. Hyman, J.M., E.A. Stanley (1988) A risk base model for the spread of the AIDS virus. Math. Biosciences 90 415–473. 69. Hyman, J.M. and E.A. Stanley (1989) The Effects of Social Mixing Patterns on the Spread of AIDS, Mathematical Approaches to Problems in Resource Management and Epidemiology,(Ithaca, NY, 1987), 190–219, Lecture Notes in Biomathematics, 81, C. Castillo-Chávez, S. A. Levin, and C. A. Shoemaker (Eds.), Springer, Berlin. 70. Isham, V. (1989) Estimation of the incidence of HIV infection, Phil. Trans. Roy. Soc. Lond. B, 325: 113–121. 71. Kaplan, E.H. What Are the Risks of Risky Sex?, Operations Research, 1989. 72. Kingsley, R. A., R. Kaslow, C.R. Jr Rinaldo, K. Detre, N. Odaka, M. VanRaden, R. Detels, B.F. Polk, J. Chimel, S.F. Kersey, D. Ostrow, B. Visscher (1987) Risk factors for seroconversion to human immunodeficiency virus among male homosexuals, Lancet 1, 345– 348. 73. Kirschner, D. (1999) Dynamics of co-infection with M. tuberculosis and HIV-1, Theor. Pop. Biol., 55: 94–109. 74. Koelle, K., S. Cobey, B. Grenfell, M. Pascual (2006) Epochal evolution shapes the phylodynamics of interpandemic influenza A (H3N2) in Humans Science 314: 1898–1903. 75. Koopman, J, C.P. Simon, J.A. Jacquez, J. Joseph, L. Sattenspiel and T Park (1988) Sexual partner selectiveness effects on homosexual HIV transmission dynamics. Journal of AIDS 1: 486–504. 76. Lagakos, S.W., L. M. Barraj, and V. de Gruttola (1988) Nonparametric analysis of truncated survival data, with applications to AIDS, Biometrika, 75: 515–523. 77. Lange, J. M. A., Paul, D. A., Huisman, H. G., De Wolf, F., Van den Berg, H., Roe!, C. A., Danner, S. A., Van der Noordaa, J., Goudsmit, J. Persistent HIV antigenaemia and decline of HIV core antibodies associated with transition to AIDS . Brit. Med. J. 293, 1459–1462 (1986).

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Chapter 9

Models for Influenza

9.1 Introduction to Influenza Models Influenza causes more morbidity and more mortality than all other respiratory diseases together. There are annual seasonal epidemics that cause about 500, 000 deaths worldwide each year. During the twentieth century there were three influenza pandemics. The World Health Organization estimates that there were 40, 000, 000– 50, 000, 000 deaths worldwide in the 1918 pandemic, 2, 000, 000 deaths worldwide in the 1957 pandemic, and 1, 000, 000 deaths worldwide in the 1968 pandemic. There has been concern since 2005 that the H5N1 strain of avian influenza could develop into a strain that can be transmitted readily from human to human and develop into another pandemic, together with a widely held belief that even if this does not occur there is likely to be an influenza pandemic in the near future. More recently, the H1N1 strain of influenza did develop into a pandemic in 2009, but fortunately its case mortality rate was low and this pandemic turned out to be much less serious than had been feared. There were 18,500 confirmed deaths, but the actual number of deaths caused by the H1N1 influenza may have been as many as 200,000. This history has aroused considerable interest in modeling both the spread of influenza and comparison of the results of possible management strategies. Vaccines are available for annual seasonal epidemics. Influenza strains mutate rapidly, and each year a judgment is made of which strains of influenza are most likely to invade. A vaccine is distributed that protects against the three strains considered most dangerous. However, if a strain radically different from previously known strains arrives, vaccine provides little or no protection and there is danger of a pandemic. As it would take at least 6 months to develop a vaccine to protect against such a new strain, it would not be possible to have a vaccine ready to protect against the initial onslaught of a new pandemic strain. Attempts are underway to try to develop a more universal vaccine.

© Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_9

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Antiviral drugs are available to treat pandemic influenza, and they may have some preventive benefits as well, but such benefits are present only while antiviral treatment is continued. Various kinds of models have been used to describe influenza outbreaks. Many public health policy decisions on coping with a possible influenza pandemic are based on construction of a contact network for a population and analysis of disease spread through this network. This analysis consists of multiple stochastic simulations requiring a substantial amount of computer time. In advance of an epidemic it is not possible to know its severity, and it would be necessary to make estimates for a range of reproduction numbers. Also, model parameters for the H1N1 influenza pandemic of 2009, especially the susceptibility to infection for different age groups, were significantly different from those for seasonal epidemics. In advance of an anticipated pandemic it may be more appropriate to use simpler models until enough data are acquired to facilitate parameter estimation. Early estimation of model parameters is extremely important for coping with a serious epidemic, and one of the outcomes of the H1N1 influenza pandemic of 2009 was development of new methods, mainly based on network models, to achieve this. Our approach is to begin with simple models and to add more structure later as more information is obtained. When an epidemic does begin, plans for management strategies need to be very detailed, and use of the simple models we describe here should be restricted to advance planning and broad understanding. We begin this chapter by developing a simple compartmental influenza transmission model and then augmenting it to include both pre-epidemic vaccination and treatment during an epidemic. We will also describe some ways in which the model can be modified to be more realistic, though more complicated. The development follows the treatment in [6, 7]. We will describe the models and the results of their analyses, but omit proofs in order to focus attention on the applications of the models. Many of the results may be found in earlier chapters.

9.2 A Basic Influenza Model Since influenza epidemics usually come and go in a time period of several months, we do not include demographic effects (births and natural deaths) in our model. Our starting point is the simple SI R model. Two aspects of influenza that are easily added are that there is an incubation period between infection and the appearance of symptoms, and that a significant fraction of people who are infected never develop symptoms but go through an asymptomatic period, during which they have some infectivity, and then recover and go to the removed compartment [35]. Thus a model should contain the compartments S (susceptible), L (latent), I (infective), A (asymptomatic), and R (removed). Specifically, we make the following assumptions: 1. There is a small number I0 of initial infectives in a population of constant total size N.

9.2 A Basic Influenza Model

313

2. The number of contacts in unit time per individual is a constant multiple β of total population size N. 3. Latent members (L) are not infective. 4. A fraction p of latent members proceed to the infective compartment at rate κ, while the remainder goes directly to an asymptomatic infective compartment (A), also at rate κ. 5. There are no disease deaths; infectives (I ) recover and leave the infective compartment at rate α, and go to the removed compartment (R). 6. Asymptomatics have infectivity reduced by a factor δ, and go to the removed compartment at rate η. These assumptions lead to the model S ′ = −Sβ(I + δA) L′ = Sβ(I + δA) − κL I ′ = pκL − αI A′ = (1 − p)κL − ηA R ′ = αI + ηA

(9.1)

with initial conditions S(0) = S0 ,

L(0) = 0,

I (0) = I0 ,

A(0) = 0,

R(0) = 0,

N = S0 + I0 .

In analyzing this model we may remove one variable since N = S + L + I + A + R is constant. It is usually convenient to remove the variable R. It is possible to show that the model (9.1) is properly posed in the sense that all variables remain non-negative for 0 ≤ t < ∞. A flow diagram for the model (9.1) is shown in Fig. 9.1. The model (9.1) is the simplest possible description for influenza having the property that there are asymptomatic infections. The question that should be in the back of our minds is whether it is a sufficiently accurate description for its predictions to be useful. The model (9.1), like the other models that we will introduce later, consists of a system of ordinary differential equations. The number of susceptibles in the population tends to a limit S∞ as t → ∞ and the final size relation may be used to find this limit without the need to solve the system of differential equations. It is more realistic to assume saturation of contacts and that β is a function of the total population size N. In general, the final size relation is an inequality. If there are no disease deaths, N is constant and β is constant even with saturation of contacts. If the disease death rate is small, it appears that the final size relation is very close to an equality and it is reasonable to assume that β is constant and use the final size relation as an equation to solve for S∞ . We may use the next generation matrix approach of [48] to calculate the basic reproduction number  p δ(1 − p) . + R0 = βN α η 

(9.2)

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9 Models for Influenza

I S

L

R A

Fig. 9.1 Diagram for the basic influenza model (9.1)

A biological interpretation of this basic reproduction number is that a latent member introduced into a population of N susceptibles becomes infective with probability p, in which case he or she causes βN/α infections during an infective period of length 1/α, or becomes asymptomatic with probability 1 − p, in which case he or she causes δβN/η infections during an asymptomatic period of length 1/η. The final size relation is given by log

  S0 S0 . = R0 1 − S∞ N

(9.3)

A very general form of the final size relation that is applicable to each of the models in this chapter is derived in Sect. 4.5. The final size relation shows that S∞ > 0. This means that some members of the population are not infected during the epidemic. The size of the epidemic, the number of (clinical) cases of influenza during the epidemic, is I0 + S0 − S∞ = N − S∞ , and the number of symptomatic cases is I0 + p(S0 − S∞ ). If there are disease deaths, with a disease survival rate f and a disease death rate (1− f ) among infectives (assuming no disease deaths of asymptomatics), the number of disease deaths is I0 + (1 − f )p(S0 − S∞ ). While mathematicians view the basic reproduction number as central in studying epidemiological models, epidemiologists may be more concerned with the attack rate, as this may be measured directly. For influenza, where there are asymptomatic

9.2 A Basic Influenza Model

315

cases, there are two attack rates. One is the clinical attack rate, which is the fraction of the population that becomes infected, defined as 1−

S∞ . N

There is also the symptomatic attack rate, defined as the fraction of the population that develops disease symptoms, defined as   S∞ p 1− . N The attack rates and the basic reproduction number are connected through the final size relation (9.3). If we know the parameters of the model we can calculate R0 from (9.2) and then solve for S∞ from (9.3). We apply the model (9.1) using parameters appropriate for the 1957 influenza pandemic as suggested by [35]. The latent period is approximately 1.9 days and the infective period is approximately 4.1 days, so that κ=

1 = 0.526, 1.9

α=η=

1 = 0.244. 4.1

We also take p = 2/3,

δ = 0.5,

f = 0.98.

As in [35] we consider a population of 2000 members, of whom 12 are infective initially. In [35] a symptomatic attack rate was assumed for each of the four age groups, and the average symptomatic attack rate for the entire population was 0.326. This implies S∞ = 1022. Then we obtain R0 = 1.37 from (9.3). Now, we use this in (9.2) to calculate βN = 0.402. We will use these data as baseline values to estimate the effect that control measures might have had. The number of clinical cases is 978 (including the initial 12), the number of symptomatic cases is 664, again including the original 12, and the number of disease deaths is approximately 13. The model (9.1) can be adapted to describe management strategies for both annual seasonal epidemics and pandemics.

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9.2.1 Vaccination To cope with annual seasonal influenza epidemics, there is a program of vaccination before the “flu” season begins. Each year a vaccine is produced aimed at protecting against the three influenza strains considered most dangerous for the coming season. We formulate a model to add vaccination to the model described by (9.1) under the assumption that vaccination reduces susceptibility (the probability of infection if a contact with an infected member of the population is made). In addition, if we assume that vaccinated members who develop infection are less likely to transmit infection, more likely not to develop symptoms, and are likely to recover more rapidly than unvaccinated members. These assumptions require us to introduce additional compartments into the model to follow treated members of the population through the stages of infection. We use the classes S, L, I, A, R as before and introduce ST , the class of treated susceptibles, LT , the class of treated latent members, IT , the class of treated infectives, and AT , the class of treated asymptomatics. In addition to the assumptions made in formulating the model (9.1) we also assume • A fraction γ of the population is vaccinated before a disease outbreak and vaccinated members have susceptibility to infection reduced by a factor σS . • There are decreases σI and σA , respectively, in infectivity in IT , and AT ; it is reasonable to assume σI < 1,

σA < 1.

• The rates of departure from LT , IT , and AT are κT , αT , and ηT , respectively. It is reasonable to assume κ ≤ κT ,

α ≤ αT ,

η ≤ ηT .

• The fractions of members recovering from disease when they leave I and IT are f and fT , respectively. It is reasonable to assume f ≤ fT . In our analysis we will take f = fT = 1 (no disease deaths). • Vaccination decreases the fraction of latent members who will develop symptoms by a factor τ , with 0 ≤ τ ≤ 1. For convenience we introduce the notation Q = I + δA + σI IT + δσA AT . The resulting model is S ′ = −SβQ

ST′ = −σS ST βQ

L′ = SβQ − κL

(9.4)

9.2 A Basic Influenza Model

317

Fig. 9.2 Transition diagram corresponding to the vaccination model (9.5)

L′T = σS ST βQ − κT LT I ′ = pκL − αI

IT′ ′

(9.5)

= τpκT LT − αT IT

A = (1 − p)κL − ηA

A′T = (1 − pτ )κT LT − ηT AT

R ′ = αI + αT IT + ηA + ηT AT .

The initial conditions are S(0) = (1 − γ )S0 , ST (0) = γ S0 , I (0) = I0 , L(0) = LT (0) = IT (0) = A(0) = AT (0) = 0

N = S0 + I0 ,

corresponding to pre-epidemic treatment of a fraction γ of the population. A flow diagram for the model (9.5) is shown in Fig. 9.2. Since the infection now is beginning in a population which is not fully susceptible, we speak of the control reproduction number Rc rather than the basic reproduction number. A computation using the next generation matrix leads to the control reproduction number Rc = (1 − γ )Ru + γ Rv , with δ(1 − p)  = R0 , α η  pτ σI δ(1 − pτ )σA  Rv = σS Nβ + . αT ηT Ru = Nβ

p

+

(9.6)

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9 Models for Influenza

Then Ru is a reproduction number for unvaccinated people and Rv is a reproduction number for vaccinated people. There is a pair of final size relations for the two final sizes S(∞) and ST (∞) in terms of the group sizes Nu = (1 − γ )N, Nv = γ N, namely     ST (∞) S∞ (1 − γ )S0 + Rv 1 − , = Ru 1 − S∞ Nu Nv     S∞ γ S0 ST (∞) = σS Ru 1 − + Rv 1 − . log ST (∞) Nu Nv

log

(9.7)

The number of symptomatic disease cases is I0 + p[(1 − γ )S0 − S(∞)] + pτ [γ S0 − ST (∞)], and the number of disease deaths is (1 − f )[I0 + p(1 − γ )S0 − S(∞)] + (1 − fT )pτ [γ S0 − ST (∞)]. These may be calculated with the aid of (9.7). By control of the epidemic we mean vaccinating enough people (i.e., taking γ large enough) to make Rc < 1. We use the following parameters, suggested in [35] and [22], σS = 0.3,

σI = σA = 0.2,

κT = 0.526,

αT = ηT = 0.323,

τ = 0.4.

With these parameter values, Ru = 1.373,

Rv = 0.047.

In order to make Rc = 1, we need to take γ = 0.28. This is the fraction of the population that needs to be vaccinated to head off an epidemic. We may solve the pair of final size equations with S(0) = (1 − γ )S0 , ST (0) = γ S0 for S(∞), ST (∞) for different values of γ . We do this for the parameter values suggested above and we obtain the results shown in Table 9.1, giving the treatment fraction γ , the number of untreated susceptibles S(∞) at the end of the epidemic, the number of treated susceptibles ST (∞) at the end of the epidemic, the number of treated cases of influenza I0 + p[(1 − γ )S0 − S(∞)], and the number of untreated cases (γ S0 − ST (∞)]. The results indicate the benefits of pre-epidemic vaccination of even a small fraction of the population in reducing the number of influenza cases. They also demonstrate the advantage of vaccination to an individual. The attack rate in the vaccinated portion of the population is much less than the attack rate in the unvaccinated portion of the population.

9.3 Antiviral Treatment

319

Table 9.1 Effect of vaccination Fraction treated 0 0.05 0.1 0.15 0.2 0.25 0.3

S∞ 1015 1079 1149 1224 1305 1395 1391

ST (∞) 0 84 174 271 375 487 596

Untreated cases 660 552 439 323 201 76 13

Treated cases 0 4 7 7 6 3 0

9.3 Antiviral Treatment If no vaccine is available for a strain of influenza it would be possible to use an antiviral treatment. However, antiviral treatment affords protection only while the treatment is continued and is more expensive. In addition, antivirals are in short supply and expensive, and treatment of enough of the population to control an anticipated epidemic may not be feasible. A policy of treatment aimed particularly at people who have been infected or who have been in contact with infectives once a disease outbreak has begun may be a more appropriate approach. This requires a model with treatment rates for latent, infective, and asymptomatically infected members of the population that we construct building on the structure used for vaccination in (9.5). Antiviral drugs have effects similar to vaccines in decreasing susceptibility to infection and decreasing infectivity, likelihood of developing symptoms, and length of infective period in case of infection. However, they are likely to be less effective than a well-matched vaccine, especially in the reduction of susceptibility. Treatment may be given to diagnosed infectives. In addition, one may treat contacts of infectives who are thought to have been infected. This is modeled by treating latent members. In practice, some of those identified by contact tracing and treated would actually be susceptibles, but we neglect this in the model. Although we have allowed treatment of asymptomatics in the model, this is unlikely to be done, and we will describe the results of the model under the assumption ϕA = θA = 0. However, for generality we retain the possibility of antiviral treatment of asymptomatics in the model. If treatment is given only to infectives, the compartments LT , AT are empty and may be omitted from the model. We add to the model (9.5) antiviral treatment of latent, infective, and asymptomatically infected members of the population, but we do not assume an initial treated class. In addition to the assumptions made earlier we also assume • There is a treatment rate ϕL in L and a rate θL of relapse from LT to L, a treatment rate ϕI in I and a rate θI of relapse from IT to I , and a treatment rate ϕA in A and a rate θA of relapse from AT to A.

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9 Models for Influenza

The resulting model is S ′ = −βSQ L′ = βSQ − κL − ϕL L + θL LT L′T = −κT LT + ϕL L − θL LT I ′ = pκL − αI − ϕI I + θI IT IT′ = pτ κT LT − αT IT + ϕI I − θI IT A′ = (1 − p)κL − ηA − ϕA A + θA AT A′T = (1 − pτ )κT LT − ηT AT + ϕA A − θA AT N ′ = −(1 − f )αI − (1 − fT )αT IT ,

(9.8)

with Q as in (9.4). The initial conditions are S(0) = S0 , I (0) = I0 , L(0) = LT (0) = IT (0) = A(0) = AT (0) = 0, N = S0 +I0 . A flow diagram for the model (9.8) is shown in Fig. 9.3. The calculation of Rc for the antiviral treatment model (9.8) is more complicated than for models considered previously, but it is possible to show that Rc = RI +RA with   Nβ (αT + θI + σI ϕI )pκ(κT + θL ) + (θI + σI (α + ϕI ))pτ κT φL RI = ΔI ΔL   δNβ (1 − p)κ(κT + θL ) σA (1 − pτ )κT ϕL RA = , (9.9) + ΔL η ηT

Fig. 9.3 Treatment model (9.8)

9.3 Antiviral Treatment

321

where ΔI = (α + ϕI )(αT + θI ) − θI ϕI ΔL = (κ + ϕL )(κT + θL ) − θL ϕL . The final size equation is   S∞ βI0 (αT + θI + σI ϕI ) S0 + = Rc 1 − . log S∞ S0 ΔI

(9.10)

The number of people treated is 

∞

0

[ϕL L(t) + ϕI I (t)]dt

and the number of disease cases is  ∞ I0 + [pκL(t) + pτ κT LT ]dt 0

which can be evaluated in terms of the parameters of the model. The number of people treated and the number of disease cases are constant multiples of S0 − S∞ plus constant multiples of the (presumably small) number of initial infectives. Since the expressions in terms of S0 − S∞ and I0 are more complicated than in the cases of no treatment or vaccination, we have chosen to give these numbers in terms of integrals. There is an important consequence of the calculation of the number of disease cases and treatments that is not at all obvious. If Rc is close to 1 or ≤ 1, S0 − S∞ depends very sensitively on changes in I0 . For example, in a population of 2000 with R0 = 1.5, a change in I0 from 1 to 2 multiplies S0 − S∞ and therefore treatments and cases by 1.4 and a change in I0 from 1 to 5 multiplies S0 − S∞ and therefore treatments and cases by 3. Thus, numerical predictions in themselves are of little value. However, comparison of different strategies is valid, and the model indicates the importance of early action while the number of infectives is small. In the special case that treatment is applied only to infectives, Rc is given by the simpler expression  p(αT + θI + σI ϕI ) δ(1 − p) . + Rc = Nβ ΔI η 

The number of disease cases is I0 + p(S0 − S∞ ), and the number of people treated is ϕI (αT + θI ) [I0 + p(S0 − S∞ )]. ΔI

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9 Models for Influenza

Since the infective period is short, antiviral treatment would normally be applied as long as the patient remains infective. Thus we assume θI = 0, and these relations become even simpler. The control reproduction number is Rc = Nβ



 p(αT + σI ϕI ) δ(1 − p) + , αT (α + ϕI ) η

(9.11)

and the number of people treated is ϕI [I0 + p(S0 − S∞ )]. α + ϕI

(9.12)

Since the basic reproduction number of a future pandemic cannot be known in advance, it is necessary to take a range of contact rates in order to make predictions. In particular, we could compare the effectiveness in controlling the number of infections or the number of disease deaths of different strategies such as treating only infectives, treating only latent members, or treating a combination of both infective and latent members. In making such comparisons, it is important to take into account that treatment of latent members must be supplied for a longer period than for infectives. In case of a pandemic, there are also questions of whether the supply of antiviral drugs will be sufficient to carry out a given strategy. For this reason, we must calculate the number of treatments corresponding to a given treatment rate. It is possible to do this from the model; the result is a constant multiple of I0 plus a multiple of S0 − S∞ . The results of such calculations appear to indicate that treatment of diagnosed infectives is the most effective strategy [6, 7]. However, there are other considerations that would go into any policy decision. For example, a pandemic would threaten to disrupt essential services and it could be decided to use antiviral drugs prophylactically in an attempt to protect health care workers and public safety personnel. A study including this aspect based on the antiviral treatment model given here is reported in [26]. For simulations, we use the initial values S0 = 1988,

I0 = 12,

the parameters of Sect. 9.1, and for antiviral efficacy we assume σS = 0.7,

σI = σA = 0.2,

based on data reported in [50]. We simulate the model (9.8) with θI = 0, assuming that treatment continues for the duration of the infection. We assume that 80% of diagnosed infectives are treated within 1 day. Since the assumption of treatment at a constant rate ϕI implies treatment of a fraction 1 − exp(−ϕI t) after a time t, we take ϕI to satisfy 1 − e−ϕI = 0.8,

9.4 Seasonal Influenza Epidemics Table 9.2 Control by antiviral treatment

323 R0 1.37 1.49 1.71 2.39

βN 0.402 0.435 0.5 0.7

Rc 0.92 1.00 1.15 1.61

Disease cases 64 130 373 865

Treatments 56 113 314 751

or ϕI = 1.61. We use different values of βN , corresponding to different values of R0 , and use (9.10) and (9.11), obtaining the results shown in Table 9.2. We calculate from (9.11) that Rc = 1 for an attack rate of 39%. This is the critical attack rate beyond which treatment at the rate specified cannot control the pandemic.

9.4 Seasonal Influenza Epidemics There are annual influenza outbreaks, usually in the fall and winter. Currently, the annual influenza outbreak is the subtype H3N2, coexisting with the subtype H1N1. Influenza A viruses are divided into subtypes identified by two proteins on the surface of the virus, namely hemagglutinin (HA) and neuraminidase (NA). The designation H3N2 means an influenza A virus that has an HA3 protein and an NA2 protein. An influenza virus undergoes antigenic drift, rapid minor genetic variation in a currently circulating subtype. Recovery from an influenza virus confers immunity against reinfection, but antigenic drift means that a later contact with an influenza virus of the same subtype will mean only partial immunity against infection. Until now, we have studied influenza epidemics in isolation, but if there is a seasonal outbreak of a strain that has been occurring for several seasons part of the population will have some cross-immunity protecting against another infection. Models for recurring seasonal outbreaks taking this into account have been analyzed by Andreasen et al. [3, 4]. Thus we now assume that at the beginning of a seasonal influenza outbreak individuals have a level of cross-immunity depending on the most recent outbreak, up to a maximum of n seasons. We assume that cross-immunity reduces both susceptibility to infection and infectivity in case of infection. At the beginning of a seasonal outbreak, we assume that all hosts are susceptible and we let Ni denote the number of hosts whose last infection occurred i seasons ago and we include in Nn all who have never been infected. We are dividing the total population into n sub-populations (N1 , N2 , · · · , Nn ) representing the distribution of cross immunities at the start of a seasonal epidemic. The (constant) total population size is N=

n

Ni .

i=1

For simplicity, we assume that during a seasonal epidemic the model is a simple SI R model, ignoring the latent period and the existence of asymptomatic cases.

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9 Models for Influenza

Thus for each class i we have Ii infectious and Ri recovered individuals. The transmission dynamics are given by the system Si′ = −σi Si β Ii′

= σi Si β

n

τj Ij

j =1

n j =1

τj Ij − αIi

(9.13)

Ri′ = αIi . Here α is the recovery rate, β is the transmission coefficient in the absence of cross-immunity, σi is the relative susceptibility of individuals whose last infection occurred i seasons ago, and τi is the relative infectivity of individuals whose last infection occurred i seasons ago. It is reasonable to assume that cross-immunity fades with time, so that 0 ≤ σ1 ≤ σ2 ≤ · · · ≤ σn = 1,

0 ≤ τ1 ≤ τ2 ≤ · · · ≤ τn = 1.

If we define the total infectivity ϕ(t) =

n

τj Ij ,

j =1

we may rewrite the model (9.13) as Si′ = −σi βSi ϕ Ii′ = σi βSi ϕ − αIi .

(9.14)

There is a disease-free equilibrium Si = Ni , Ii = 0(i = 1, 2, · · · , n). The next generation matrix is ⎡

σ1 N1 τ1 σ2 N1 τ1 ⎢ β ⎢ σ2 N2 τ1 σ2 N2 τ2 K= ⎢ . .. α ⎣ .. . σn Nn τ1 σn Nn τ2

⎤ · · · σn N1 τ1 · · · σn N2 τ2 ⎥ ⎥ .. ⎥ . .. . . ⎦ · · · σn Nn τn

Now, if P is the diagonal matrix with diagonal entries σi Ni , 1 ≤ i ≤ n, the matrix P −1 KP similar to K has every row β/α multiplied by (σ1 τ1 N1 , σ2 τ2 N2 , · · · , σn τn Nn ) and therefore has rank 1. This implies that all but one of the eigenvalues of K are zero, and the remaining eigenvalue is equal to the trace of K. From this we conclude that the spectral radius of K is equal to the trace of K and the control reproduction number of the cross-immunity model (9.14) is

9.4 Seasonal Influenza Epidemics

325

RS =

n β σ1 τ1 Ni . α

(9.15)

i=1

If there were no cross-immunity, the basic reproduction number would be βN . α

R0 =

Because (Si + Ii )′ = −αIi , we can deduce as in Sect. 2.4 that lim Ii (t) = 0,

lim Si (t) = Si (∞),

t→∞

t→∞

and α



∞

0

Ii (t)dt = Ni − Si (∞),

using Si (0) + Ii (0) = Ni . Integration of the first equation of (9.14) gives Ni = βσi log Si (∞) = βσi



∞

ϕ(t)dt

0

n

τj

j =1



∞

Ij (t)dt

(9.16)

0

n

=

β σi τj [Nj − Sj (∞)] α j =1

= σi

n j =1

  Sj (∞) βNj 1− . τj α Nj

We can write this final size relation in terms of the single unknown Sn (∞) because from (9.14) and σn = 1 we have Si′ S′ = σi n , Si Sn and integration gives   Si (∞) Sn (∞) σi = . Ni Nn Substitution of (9.17) into (9.16) gives a relation for Sn (∞), namely

(9.17)

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9 Models for Influenza

log

    n βNj Sn (∞) σj Nn . 1− = σn τj Sn (∞) α Nn j =1

Then (9.17) gives solutions for Si (∞), i = 1, . . . , n − 1.

9.4.1 Season to Season Transition The above analysis traces the development of a seasonal epidemic. We assume that at the beginning of the next season’s epidemic, the final compartment sizes give the separation of the population into new cross-immunity groups (N1∗ , N2∗ , . . . , Nn∗ ). The individuals who were infected during the previous season’s epidemic form the ∗ . Thus new group N1∗ , and the remaining individuals from Ni move to Ni+1 N1∗ = N −

n

Si (∞)

i=1

Nj∗ = Sj −1 (∞),

j = 2, · · · , n − 1

Nn∗ = Sn−1 (∞) + Sn (∞).

This relation describes the transition from one seasonal epidemic to the next. The severity of one season’s epidemic affects the cross-immunity for the next season’s epidemic. If the seasonal epidemic 1 year is severe, then because there is more crossimmunity the next year it is reasonable to expect a less severe epidemic the next year.

9.5 Pandemic Influenza In some years, there is an immunological change that produces a new subtype. Such a change is called an antigenic shift, and is usually a result of recombination of gene segments from viruses circulating in humans with virus segments from avian viruses. Since this subtype is new, humans have little or no immunity against it, and the result often is a pandemic. For example, the H1N1 subtype emerged in the pandemic influenza of 1918 and circulated until 1957. In the pandemic of 1957 a new subtype, H2N2, emerged and replaced H1N1. In the pandemic of 1968, H3N2 replaced the prevailing subtype H2N2. In 1977, H1N1 was reintroduced and has been circulating along with H3N2 since then. A natural question is whether a new pandemic strain will replace the currently circulating subtype or coexist with it. To study this question, we must model a pandemic following a seasonal influenza outbreak and then model the next seasonal outbreak. This is necessary because there may be some cross-immunity between the seasonal and pandemic subtypes. The interplay between seasonal and pandemic influenza has been studied by Asaduzzaman et al. [8].

9.5 Pandemic Influenza

327

9.5.1 A Pandemic Outbreak We begin by assuming that there has been a seasonal influenza epidemic and that at the end of this epidemic the population of total size N is divided into cross-immunity sub-populations with respect to the seasonal epidemics of sizes (N1 , N2 , · · · , Nn ). We assume that there is a cross-immunity between the pandemic strain and the seasonal strain, for individuals in the population who have recovered from the seasonal strain in the previous n seasonal outbreaks reducing susceptibility by a factor ρ, (0 ≤ ρ ≤ 1). Thus ρ = 0 corresponds to full cross-immunity and ρ = 1 corresponds to no cross-immunity. Then the model for pandemic influenza is ˜ i I, i = 1, 2, · · · , n − 1 Si′ = −ρ βS ˜ nI Sn′ = −βS I ′ = β˜ ρ(S1 + S2 + · · · + Sn−1 ) + Sn I − αI.

(9.18)

If there were no cross-immunity, the basic reproduction number would be ˜ βN , R˜ 0 = α

(9.19)

where β˜ is the pandemic contact rate. An analysis very similar to that of Sect. 9.5 shows that the pandemic reproduction number is RP =

ρ βN ˜ ˜ n β˜  (1 − ρ)βN ρ(N1 + N2 + · · · + Nn−1 ) + Nn = + , α α α

an increasing function of ρ. If RP > 1, there will be a pandemic. We assume that the pandemic would invade if there were no cross-immunity, that is, R˜ 0 > 1. Also, again much as in Sect. 9.5, we obtain a final size relation

n   ˜ βN Nn i=1 Si (∞) = 1− log Sn (∞) α N ρ  Si (∞) Sn (∞) , i = 1, · · · , n − 1. = Ni Nn

(9.20)

This relation gives the cross-immunity distribution for the pandemic. When we model the seasonal epidemic following the pandemic, we will need this result in order to distinguish between individuals susceptible to the seasonal influenza

328

9 Models for Influenza

virus having cross-immunity to the pandemic strain and individuals without crossimmunity.

9.5.2 Seasonal Outbreaks Following a Pandemic In order to model a seasonal outbreak following a pandemic, because of crossimmunity between the seasonal and pandemic viruses, we must divide each of the subgroups (N1 , N2 , · · · , Nn ) into two parts, depending on whether the members of the group were infected by the pandemic. Now, we let Mi denote the individuals whose most recent seasonal infection was i seasons ago and who escaped the pandemic, and let M˜ i denote the individuals whose most recent seasonal infection was i seasons ago and who were infected in the pandemic. Then Ni = Mi + M˜ i . We denote by Si the number of individuals in Mi susceptible to the seasonal influenza (who escaped the pandemic influenza) and by S˜i the number of individuals in Mi susceptible to the seasonal influenza (infected by the pandemic). Taking into account the cross-immunity of individuals in Mi and again defining the total infectivity ϕ(t) =

n

τj Ij ,

j =1

we obtain the model Si′ = −σi βSi ϕ S˜ ′ = −ρσi S˜i ϕ

(9.21)

Ii′ = σi β(Si + ρ S˜i )ϕ − αIi .

The calculation of the reproduction number is again much the same as in Sect. 9.4, and yields the result R=

n β σi τi (Mi + ρ M˜ i ). α i=1

In order to evaluate this, we need to determine Mi and M˜ i from the final size relation (9.20). We have Mi = Si (∞), M˜ i = Ni − Si (∞), so that Mi + ρ M˜ i = Mi (1 − ρ) + ρNi , with Si (∞) given by (9.20). If we let x =

Sn (∞) Nn ,

then (9.20) gives

9.5 Pandemic Influenza

329

− log x =

n β˜ [N − Si (∞)] α

(9.22)

i=1

 ρ ρ ρ ˜ βN N1 Sn + N2 Sn + · · · + Nn Sn 1− = ρ α N Nn =

˜ βN [1 − x ρ ]. α

Consider the function f (x) = log x +

˜ βN [1 − x ρ ] = log x + R˜ 0 [1 − x ρ ], α

where R˜ 0 is given by (9.19). A solution x of (9.22) is epidemiologically meaningful only if x < 1; otherwise, the pandemic does not invade. The value x = 1 is always a root of (9.22); the pandemic invades if and only if there is a second root less than 1. The function f (x) is negative for x close to 0. If ρ R˜ < 1, f (x) is monotone increasing for 0 < x ≤ 1 and therefore there is no second root of (9.22) less than 1. Thus, if ρ R˜ 0 < 1, the pandemic does not invade. On the other hand, if ρ R˜ 0 > 1, f ′ (1) < 0, and there must be a second root of (9.22). The pandemic invades if and only if ρ R˜ 0 > 1. Now, the reproduction number is R=

n β σi τi [(1 − ρ)Ni x ρ + ρNi ] α i=1)

n

n

i=1

i=1

β β σi τi Ni x ρ + ρ σi τi Ni . = (1 − ρ) α α

(9.23)

If R, with x given as a function of ρ by (9.22), is greater than 1, then the pandemic and seasonal strains will coexist, but if R < 1, then the pandemic strain will replace the seasonal strain. We assume that the seasonal epidemic would take place if there were no pandemic strain present, that is, RS > 1. For ρ = 1 (no cross-immunity) we have R = RS > 1, and this means that the pandemic and seasonal strains coexist. For ρ = 1/R˜ 0 , x = 1, and

330

9 Models for Influenza

R=

n β σi τi Ni = RS > 1, α i=1

and the pandemic and seasonal strains coexist. Numerical simulations indicate that R < 1 for some values of ρ between 1/R˜ 0 and 1, which implies that for some values of ρ the pandemic strain will replace the seasonal strain.

9.6 The Influenza Pandemic of 2009 In the spring of 2009 a new strain (H1N1) of influenza A was developed, apparently first in Mexico, and spread rapidly through much of the world. Initially, it was thought that there was no resistance to this strain, although it developed later that people who were old enough to have been exposed to similar strains in the 1950s appeared to be less susceptible than younger people. It also appeared initially that this strain had a high case fatality rate, but it was learned later that since many cases were mild enough not to be reported the early data were weighted towards severe cases. The case fatality rate was actually lower than for most seasonal influenza strains. Management of the H1N1 influenza pandemic of 2009 made use of mathematical models and the experience gained from previous epidemics, but also exposed some gaps between a well-developed mathematical theory of epidemics and reallife epidemics, notably in the acquisition of reliable data early in the pandemic, understanding of spatial spread of a pandemic, and the development of multiple epidemic waves. There are important differences in the kinds of models of value to different types of scientists. There are strategic disease transmission models aimed at the understanding of broad general principles, and tactical models with the goal of helping make decisions in specific and detailed situations on short-term policies. A model that predicts how many people will become ill in an anticipated epidemic is quite different from a model that helps to identify which subgroups of a population should have highest priority for preventive vaccination with an uncertain prediction of how much vaccine will be available in a given time frame.

9.6.1 A Tactical Influenza Model There was a second wave of infections in the 2009 H1N1 influenza pandemic, just as in several previous influenza pandemics. As soon as possible, during the first wave, work began on the development of a vaccine matched to the virus to be used to combat the expected second wave [41]. Since a vaccine requires at least 6 months

9.6 The Influenza Pandemic of 2009

331

for development, and the second wave could begin within 6 months of the first wave, there was an urgent need to prepare a vaccine distribution strategy. In this section, we describe the model of [18] as an example of a tactical model. This model was used by the BC Center for Disease Control in making planning decisions for vaccination distribution during the second wave of the 2009 H1N1 influenza pandemic. Because of the need to make such decisions rapidly, the development and application of the model was more urgent than the writing of the paper that described the work, and use of the results preceded the write-up. In a pandemic, there is still much to be done after the model has been developed and applied, and frequently there is no time to explore basic theoretical questions that may arise in the study. Ideally, these questions can be studied in more depth after the urgency of coping with a pandemic has passed, and this is an opportunity to return to more general strategic models. Infection rates in an influenza epidemic depend strongly on the (demographic) age of the individuals in contact. This dependence of transmission and infection rates on age (the age profile) may vary significantly between locations and from year to year for seasonal epidemics. Also, the age profile in a pandemic is probably unlike that for seasonal epidemics. For this reason, real-time planning during an epidemic must make use of surveillance data obtained as soon as possible after the epidemic has begun. The data gathered during the first wave in the spring of 2009 were used to project the age profile to be expected in the second wave. A compartmental model with 6 compartments in each of the 40 population subgroups, covering 8 age classes and 5 activity levels using this age profile and existing estimates of the Vancouver contact network structure, was developed to project the results of different vaccination strategies. The analysis of the model was carried out using numerical simulations, because comparisons of different strategies, including numerical estimates of disease cases and vaccine quantities required, were needed quickly and because general theoretical analysis of this high-dimensional system would have been too complicated. The numerical simulations indicated that a good estimate of the epidemic peak would be essential for making policy decisions on vaccination strategies. In a pandemic situation, delays in vaccine production may mean that vaccination cannot be started until an epidemic is already underway, and the model suggested that an early start to vaccine distribution makes a big difference in the effectiveness of the vaccination program. This raises a general theoretical question of the relation between the starting time of vaccination and the effectiveness of the vaccination program. Future theoretical study of models of this type would be very useful for planning vaccination strategies for epidemics in the future.

9.6.2 Multiple Epidemic Waves During the 1918 “Spanish flu” pandemic, North America and much of Western Europe experienced two waves of infection with the second wave more severe than

332

9 Models for Influenza

the first [20, 21]. Unlike seasonal influenza epidemics, which occur at predictable times each year, influenza pandemics are often shifted slightly from the usual “influenza season” and have multiple waves of varying severity. This suggests a modeling question that does not arise with seasonal influenza. During a first wave of a pandemic it should be possible to isolate virus samples and begin development of a vaccine matched to the virus in time to allow vaccination against the virus that could help to manage a second wave. One question that arises is prediction of the timing of a second wave. This is not yet possible because there is not yet a satisfactory explanation of the causes of a second wave. One suggestion that has been made is that transmissibility of virus varies seasonally, and this has been used to try to predict whether an endemic disease will exhibit seasonal outbreaks [44, 45]. The same approach can be used to formulate an SI R epidemic model with a periodic contact rate that can exhibit two epidemic waves [12]. However, the behavior of such a model depends strongly on the timing of the epidemic. If we assume that the contact rate is highest in the winter, lowest in the summer, and varies sinusoidally with a period of 1 year, we could use a simple SI R model with a variable contact rate. With N as the (constant) total population size, this suggests a model SI S ′ = −β(t) N SI − αI, I ′ = β(t) N

(9.24)

where    π(t + t0 ) , β(t) = β 1 + c cos 180 with parameter values α = 0.25,

β = 0.45,

c = 0.45,

N = 1000,

S0 = 999,

I0 = 1,

t0 = 85, 90, 95.

The choice of t0 determines where in the oscillation of β(t) the epidemic begins, because at the start of the epidemic (t = 0)    π t0 . β(0) = β 1 + c cos 180

Thus t0 is the number of days after the maximum transmissibility that the epidemic begins. By numerical integration of (9.24), we obtain the following three epidemic curves (giving the number of infective individuals as a function of time), with the same parameters except that by varying t0 the starting date of the epidemic is moved 5 days later from one curve to the next.

9.6 The Influenza Pandemic of 2009

333

Fig. 9.4 Top: A one-wave epidemic curve, t0 = 85. Middle: A two-wave epidemic curve, first wave more severe, t0 = 90. Bottom: A two-wave epidemic curve, second wave more severe, t0 = 95

An interpretation of these curves is that if the epidemic begins when the contact rate is decreasing and is close to its minimum value and the contact rate is relatively small over the course of the epidemic, the wave may end while there are still enough susceptibles to support a second wave when the contact rate increases, as in Fig. 9.4 (middle and bottom plots). However, if the epidemic begins earlier it may continue until enough individuals are infected that a second wave cannot be supported even when the contact rate becomes large, as in the top plot of Fig. 9.4. Simulations indicate that for the model (9.24) with the parameter values used here there is a small window of starting times corresponding to the interval 90 ≤ t0 ≤ 110 for which a second wave is possible. The nature of the epidemic curves indicates that the behavior depends critically on timing and this means that such a model is not suitable for precise predictions. Note that there may be a single wave or two waves, and if there are two waves either one may be more severe. Not only does the prediction depend on the timing of the introduction of infection, but this timing is stochastic, and by its very nature unpredictable. It depends on mutations of the virus and importations of new cases. We have here a strategic model that predicts the possibility of a second wave, but this is not sufficient to be confident about using it to advise policy, because we do not have enough evidence to be sure about the cause of a second wave. Before

334

9 Models for Influenza

we could make predictions we need more evidence about factors that could cause a second wave and more detailed tactical models, including sensitivity analysis. Another factor that has been suggested as a possible explanation for a second epidemic wave is coinfection with other respiratory infections that might increase susceptibility, and some experimental data would be needed that might confirm or deny that either seasonal variation in transmission or coinfection, or both, could explain multiple pandemic waves. In the H1N1 influenza pandemic of 2009, there were some concerns about the development of drug resistance as a consequence of antiviral treatment. While this did not appear to have widespread consequences, in a more severe disease outbreak in which more patients receive antiviral treatment there could be major effects. The modeling of drug resistance effects in an influenza pandemic has begun [1, 5, 36, 42, 43], but much more needs to be understood. A full analysis of the development of drug resistance will require nested models including immunological in-host aspects as well as effects on the population level. Data from the 1918 pandemic indicate clearly that behavioral response, both individual and public health measures, had significant effects on the outcome of the epidemic [9]. Incorporating behavioral responses is a new aspect of epidemic modeling, and there is much to be learned about the factors that influence behavioral responses when a disease outbreak begins.

9.6.3 Parameter Estimation and Forecast of the Fall Wave The model predictions included in this section are presented in [47]. With the recognition of a new, potentially pandemic strain of influenza A(H1N1) in April 2009, the laboratories at the US CDC and the World Health Organization (WHO) dramatically increased their testing activity from week 17 onwards (week ending 2 May 2009), as can be seen in Fig. 9.5. In this analysis, we use the extrapolation of a model fitted to the confirmed influenza A(H1N1)v case counts during summer 2009 to predict the behavior of the pandemic during autumn 2009. The following model with seasonally forced transmission is used: SI S ′ = −β(t) N SI − αI, I ′ = β(t) N

(9.25)

where N = 305,000,000 denote the total population in the United States (US). The R equation is omitted as it can be determined by R = N − S − I . The transmission rate is chosen to be β(t) = β0 + β1 cos(π t/180),

(9.26)

9.6 The Influenza Pandemic of 2009

335

Fig. 9.5 Influenza-positive tests reported to the US CDC by US WHO/NERVSS-collaborating laboratories, national summary, United States, 2009 until 26 September

where β0 and β1 are constants to be estimated from data. Assume that I (t0 ) = 1, where t0 is the initial time, which is another parameter to be estimated from data. The parameter α is chosen so that the infective period is 1/α = 3 days. The three parameters (β0 , β1 , t0 ) were estimated using the data shown in Fig. 9.5 on influenza-positive tests reported to the US CDC by US WHO/NREVSScollaborating laboratories, national summary, United States, 2009 until 26 September. To avoid bias due to increased testing for H1N1 around week 16, only data from week 21 to 33 (from 24 May to 22 August 2009) were used. From the past experience with influenza, the lower and upper bounds were chosen to be β0 ∈ (0.92α, 2.52α) and β1 ∈ (0.05α, 0.8α), and t0 ∈ (−8, 10) (weeks relative to the beginning of 2009). The best estimates were determined by fitting the model to data, and the parameter values that provided the best Pearson chi-square statistics are listed in Table 9.3. Using the estimated parameter values, forecast of the time and size of the fall wave was produced by simulating the model under two scenarios: one is without vaccination and the other included the planned CDC vaccination program, which would begin with six to seven million doses being delivered by the end of the first full week in October (week 40), with 10–20 million doses being delivered

336 Table 9.3 Estimates of parameters for the 2009 H1N1 model

9 Models for Influenza Parameter β0 β1 t0

Value 1.56 0.54 24 February

95% confidence interval (1.43, 1.77) (0.39, 0.54) (8 February, 7 March)

Fig. 9.6 Prediction for the 2009 H1N1 pandemic in the US using model (9.25)

weekly thereafter. It is assumed that for healthy adults, full immunity to H1N1 influenza is achieved about 2 weeks after vaccination with one dose of vaccine [30, 37]. For simulations of the model when vaccinations are incorporated, the number of susceptible individuals in the simulations was decreased according to the appropriate proportion of immune due to vaccination. The simulation outcomes are presented in Fig. 9.6. In Fig. 9.6, the curves associated with the darker and lighter areas correspond to the cases of with and without vaccination. The model predicts that in the absence of vaccine, the peak wave of infection will occur near the end of October in week 42 (95% CI: week 39, 43). By the end of 2009, the model predicts that a total of 63% of the population will have been infected (95% CI: 57%, 70%). For the case when the planned vaccination program is considered, the model results suggest that a relative reduction of about 6% in the total number of people infected with influenza A(H1N1)v virus by the end of the year 2009 (95% CI: 1%, 17%). The most striking feature of the model is that it accurately predicted the peak time of the pandemic. According to CDC 2009 H1N1 confirmed case count data (see [17]), the peak of the fall wave was reached at the end of October (which is between weeks 42 and 43, see the left-hand plot in Fig. 9.7), which is consistent with

9.7 *An SIQR Model with Multiple Strains with Cross-Immunity

337

Fig. 9.7 The left figure illustrates the CDC 2009 confirmed H1N1 count data (the error bars represent variations calculated by the proposers that account for US regional variability in the timing of the pandemic). The right figure shows predictions by our model in [47]

our model result. It is worth noting that the model used in the analysis is a simple SIR model with a seasonally forced infection rate. Although further examinations are certainly needed to study the applicability of the modeling approach to general scenarios, the model results in this analysis demonstrated clearly the advantage and capability of mathematical models in understanding disease dynamics.

9.7 *An SIQR Model with Multiple Strains with Cross-Immunity It was during the Black Death in the twentieth century in Venice that a system of quarantine was first put in operation. It demanded ships to lay at anchor for “40 days” (in Latin “quaranti giorni”) before sailors and guests could come on land. Quarantine is often thought of as a policy that separates individuals who may have been exposed to a contagious agent regardless of symptoms. Isolation is, generally speaking, considered a severe form of quarantine, often put in place in response to high morbidity and mortality. Quarantine was used extensively in the treatment and control of tuberculosis in Europe first and later, at end of the nineteenth century, in the USA. The emergence of SARS in 2003 re-instated the world’s interests in the concepts of Isolation and Quarantine (Q & I), the only initial methods available of disease control [11, 19]. The concepts of Q & I have multiple working meanings and uses. Hence, selecting a definition depends on the disease, the suspected level of risk that it poses to others, the means and modes of transmission, and the system’s knowledge and experience with the infectious agent. Regardless of the definition used, one of the challenges associated with Q & I strategies is that there is hardly any reliable assessments of their population level efficacy. There are no effective

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9 Models for Influenza

quantitative frameworks that account for direct and indirect economic losses and/or the costs associated with the implementation of Q & I strategies (but see [14, 27– 29, 33, 34, 38, 40]). Critical to any method of assessment comes from the fact that dynamic models that include the Q & I classes disease must be prepared to account for their sometimes destabilizing impact on the disease dynamics (sustained oscillations). The introduction of Q & I classes can generate the kind of dynamics where assessing the effectiveness of interventions may be difficult. Feng et al. showed in [24, 25], for example, that the incorporation of a quarantine or isolation class (Q) was enough to destabilize the unique disease endemic equilibrium in a susceptible–infected– recovered (SIR) model. These results were confirmed by Hethcote and collaborators [31] using alternative SIQR modeling frameworks. Tracing exposed individuals, assuming that a test exists that determines if an individual is infected or not, and their contacts, would make it possible to quarantine or isolate diagnosed infectives, a first step towards assessing the impact of Q & I [15]. Models are used to address questions like: What impact will placing a fixed proportion of individuals living in the “neighborhood” of an index case in quarantine have on disease control? We observe that when large numbers are involved, the costs and challenges become immense. Should isolated individuals be kept at their homes or moved to designated quarantine facilities? In [39] a two-strain influenza model is studied to investigate competitive outcomes (mediated by cross-immunity) that result from the interactions between two strains of influenza A in a population where sick individuals may be quarantined or isolated. The inclusion of a quarantined class makes the model analysis much more challenging due to the presence of sustained oscillations and possible other bifurcations. To make the analysis more transparent, an SIQR model with a single strain is presented first.

9.7.1 *An SIQR Model with a Single Infectious Class In this section, results in [24] and [25] are revisited in order to highlight the impact of the inclusion of quarantine and/or isolation classes in generating sustained periodic solutions from SIQR epidemic models. Let S(t), I (t), Q(t), and R(t) denote the susceptible, infective, quarantined, and recovered classes, respectively, and let N = S+I +Q+R denote the total population size. Model parameters are: μ, the per-capita rates for birth and death; γ , the rate at which infected individuals are isolated (quarantined); δ, recovery rate; and β, the transmission rate. The SIQR model can be formulated as follows:

9.7 *An SIQR Model with Multiple Strains with Cross-Immunity

339

I dS = μN − βS − μS, dt N −Q dI I = βS − (μ + γ )I, dt N −Q dQ = γ I − (μ + δ)Q, dt dR = δQ − μR dt

(9.27)

with appropriate initial conditions. What makes the above model “distinct” is that the incidence rate now accounts for the possibility that a large number of individuals (those in the Q class) do not participate (by request, mandate, or personal decision) in the transmission process. I Hence, the “random mixing” of infected proportion with other individuals is N −Q

rather than NI . The basic reproduction number for the above SIQR model is R0 =

β . μ+γ

(9.28)

It was shown in [25] that the disease-free equilibrium is globally asymptotically stable when R0 < 1. However, for R0 > 1, the behavior of model solutions is very different from the standard SIR model. That is, the unique endemic equilibrium, denoted by E ∗ , can become unstable due to the appearance of stable periodic solution via a Hopf bifurcation [32]. Notice from (9.28) that R0 is independent of the isolation period 1/δ. A bifurcation analysis using δ as the bifurcation parameter shows that E ∗ is asymptotically stable when the quarantine period is either very large or very small. To facilitate the analysis, consider the following equivalent system with re-scaled parameters:   I +R dI = 1− I − (ν + θ )I, dτ N −Q dQ = θ I − (ν + α)Q, dτ dR = αQ − νR, dτ

(9.29)

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9 Models for Influenza

where τ = βt and ν=

μ , β

θ=

γ , β

α=

δ . β

The S equation has been eliminated as N is constant and S = N − I − Q − R. Note that the scaled parameter representing the isolation period is α. Based on the system (9.29), the following result on the possibility of periodic solutions via Hopf bifurcation was established in [25]: Theorem 9.1 There is a function, αc (ν), defined for small ν > 0, αc (ν) = θ 2 (1 − θ ) + O(ν 1/2 ), such that there are two critical values αc1 and αc2 with the following properties: (a) The endemic equilibrium is locally asymptotically stable if 1/α < 1/αc1 (ν) or 1/α > 1/αc2 (ν) and unstable if 1/αc1 (ν) < 1/α < 1/αc2 (ν), as long as 1/α is close to the critical values. (b) Hopf bifurcations occur at α = αci (ν) (i = 1, 2), leading to stable periodic solutions near the bifurcation points. Further, at the bifurcation point corresponding to αc1 , the length of periods can be approximated by the formula T ≈

2π 2π ≈ 1/2 1/2 (1 − θ ) ν (θy ∗ )1/2 ,

where y ∗ = I ∗ /(N − Q∗ ) is the proportion of infectious individuals at the endemic equilibrium scaled by the active population, N − Q. With parameter values relevant to scarlet fever, numerical simulations of the model confirm the occurrence of two Hopf bifurcation points, which are illustrated in Fig. 9.8. The right plot in Fig. 9.8 shows two Hopf bifurcation points (labeled by HB) occurring for values of the average quarantine periods in two distinct ranges near 1/αci (i = 1, 2). The plot on the left shows the enlarged portion near the lower bifurcation point 1/αc1 , with the curves labeled with SP representing the maximum and minimum of the stable periodic solutions. The solid and dashed curves labeled by SSS and USS represent stable steady state and unstable steady state (showing the I ∗ component of the endemic equilibrium E ∗ ), respectively. It was pointed out in [24, 25] that only the region near the lower critical point 1/αc1 is relevant for childhood diseases, and that the data on the length of reported isolation periods during the 1897–1978 scarlet fever epidemics in England and Wales [2] were very close to the range that supports periodic solutions for model (9.29) near 1/αc1 .

9.7 *An SIQR Model with Multiple Strains with Cross-Immunity

341

Fig. 9.8 The plot on the right is a bifurcation diagram based on numerical simulations of the system (9.29) with parameter values relevant to scarlet fever. The solid and dashed curves (labeled by SSS and USS) represent the fraction of the I components when the endemic equilibrium E ∗ is stable and unstable, respectively, depending on the value of the isolation period 1/α. The right plot shows two Hopf bifurcation points, αc1 and αc2 , labeled by HB. An enlarged version of the right HB is shown on the left. The solid curves labeled with sp illustrate the maximum and minimum of the stable periodic solutions

Several extensions of the model (9.27) have been considered including [23, 31, 49]. For example, the model in [23] reads dS I (1 − σ )Q ˆ = μN − βS − βS − μS, dt N − σQ N − σQ I (1 − σ )Q dI ˆ = βS + βS − (γ + κ + μ)I, dt N − σQ N − σQ dQ = γ I − (δ + μ)Q, dt dR = κI + δQ − μR. dt

(9.30)

In (9.30), individuals in the I class can recover without going through the Q class. In addition, the degree of effectiveness (i.e., reducing the movement of quarantined individuals) is considered through the use of the parameter σ with σ = 1 and 0 corresponding to a perfect and ineffective quarantine/isolation, respectively. It is shown in [23] that the likelihood for the existence of sustained oscillations depends on σ . Because Hopf bifurcations are local properties, and because the bifurcation diagram generated by numerical simulations in Fig. 9.8 seems to suggest that the periods of the periodic solutions for parameters away from the Hopf bifurcation points become very large, it indicates the possibility for a homoclinic bifurcation. This is explored in [51]. It was shown in [51] that a center manifold reduction of the system (9.27) at a bifurcation point has the normal form x ′ = y, y ′ = axy + bx 2 y + O(4), indicating a bifurcation of codimension greater than two. Here, x and y are variables after several changes of variables from the system (9.29). By considering an unfolding of the normal form, it was shown that a perturbed system can indeed generate a homoclinic bifurcation. This is illustrated in Fig. 9.9.

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9 Models for Influenza

Fig. 9.9 Numerical simulations of a perturbed system of the normal form reduction from the system (9.29). The four plots correspond to four sets of parameter values near the homoclinic bifurcation. As the parameter values change, the interior equilibrium E ∗ changes from stable to unstable leading to the appearance of a stable periodic solution (see plots (a) and (b)), and that a homoclinic bifurcation occurs as parameter values change from (b) to (d), in which case, a homoclinic solution exists (see (c))

The four sub-figures a–d in Fig. 9.9 demonstrate trajectories of the perturbed system for four sets of parameter values. It shows first that the stability of E ∗ switches from stable to unstable with the appearance of a stable periodic solution (see a and b). As parameter values continue to change from b to c, the system undergoes a homoclinic bifurcation, in which case, the stable periodic solution expends its magnitude and eventually disappears after reaching the saddle point (see b–d) and a homoclinic orbit appears (see c).

9.7.2 *The Case of Two Strains with Cross-Immunity The SIQR model (9.27) can be extended to include two pathogen strains. The description of the two-strain model requires the division of the population into ten different classes: susceptibles (S), infected with strain i (Ii , primary infection),

9.7 *An SIQR Model with Multiple Strains with Cross-Immunity

343

isolated with strain i (Qi ), recovered from strain i (Ri , as a result of primary infection), infected with strain i (Vi , secondary infection), given that the population had recovered from strains j = i, and recovered from both strains (W ). The population is assumed to mix randomly, except that the mixing is impacted by the process of quarantine/isolation [15, 25, 31]. Using the flow diagram in Fig. 9.10, we arrive at the model dS dt dIi dt dQi dt dRi dt dVi dt dW dt A

2

(Ii + Vi ) − μS, A i=1 (Ii + Vi ) = βi S − (μ + γi + δi )Ii , A

= Λ−

βi S

= δi Ii − (μ + αi )Qi , = γi Ii + αi Qi − βj σij Ri = βi σij Rj =

(Ij + Vj ) − μRi , A

(Ii + Vi ) − (μ + γi )Vi , A

(9.31) j = i

j = i, i, j = 1, 2

2

− μW,

2 = S + W + i=1 (Ii + Vi + Ri ), i=1 γi Vi

Fig. 9.10 Schematic diagram of the dynamics in host exposed to two co-circulating influenza strains. Λ is the rate at which individuals are born into the population, βi denotes the transmission coefficient for strain i, μ is the per-capita mortality rate, δi is the per-capita isolation rate for strain i, γi denotes the per-capita recovery rate from strain i, αi is the per-capita rate at which individuals leave the isolated class as a result of infection with strain i, and σij is the relative susceptibility to strain j for an individual that has been infected with and recovered from strain i (i = j ). σij = 0 corresponds to total cross-immunity, while σij = 1 indicates no cross-immunity between strains. The protection is said to be strong if 0 ≤ σij ≪ 1, and weak if 0 ≪ σij ≤ 1

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9 Models for Influenza

where A denotes the population of non-isolated individuals and βi S(IAi +Vi ) models the rate at which susceptibles become infected with strain i. That is, the ith (i = j ) incidence rate is assumed to be proportional to both the number of susceptibles and i) the available modified proportion of i-infectious individuals, (Ii +V A . The parameter σij is a measure of the cross-immunity provided by a prior infection with strain i to exposure with strain j (i = j ). It is assumed that σij ∈ [0, 1]. Model (1) includes the models in [13, 16]. The omission of the Q classes in earlier work precludes the possibility of sustained oscillations (see [13, 16]) in the absence of population (age) structure. System (9.31) has at least four equilibria. Analysis of the local stability of the trivial equilibrium (absence of disease) helps identify conditions under which the “flu” can invade. Hence, we first focus on establishing the conditions that make it impossible (at least locally) for both strains to invade a disease-free population, simultaneously. In the analytical results obtained here it is assumed that σ12 = σ21 = σ . The basic reproduction number for the two strains is Ri =

βi . μ + γi + δi

Using the perturbation technique based on the fact that μ is a much smaller parameter than other parameters, we can identify two curves in the (R1 , R2 ) plane that determine the stability of the non-trivial equilibria. Let f (R1 ) and g(R2 ) be two functions defined by f (R1 ) =

R1  δ2 1− 1 + σ (R1 − 1) 1 + μ+γ 2 



(9.32)

.

(9.33)

μ(μ+α1 ) (μ+γ1 )(μ+α1 )+α1 δ1

and g(R2 ) =



1 + σ (R2 − 1) 1 +

R2 

δ1 μ+γ1

1−

μ(μ+α2 ) (μ+γ2 )(μ+α1 )+α2 δ2

Let Ri∗ = Ri (i = 1, 2) evaluated at μ = 0. Then the following result holds. Theorem 9.2 There exists a function α1c (μ) defined for small μ > 0 by α1c (μ) =

δ1 1 (1 − ∗ ) + O(μ1/2 ), ∗ R1 R1

with the following properties: (i) The boundary endemic equilibrium E1 is locally asymptotically stable if R2 < f (R1 ) and α1 < α1c (μ), and unstable if R2 > f (R1 ) or α1 > α1c (μ). (ii) When R2 < f (R1 ), periodic solutions arise at α1 = α1c (μ) via Hopf bifurcation for small enough μ > 0. The period can be approximated by

9.7 *An SIQR Model with Multiple Strains with Cross-Immunity

T =

345

2π 2π ≈

. 1 |ℑω2,3 | (γ1 + δ1 )(R1∗ − 1) 2 μ1/2

Because we have focused on the symmetric case, an analogous result for the second boundary equilibrium E2 can be stated immediately. That is, the boundary endemic equilibrium E2 is locally asymptotically stable if R1 < g(R2 ) and α2 < α2c (μ). It becomes unstable if R1 > g(R2 ) or α2 > α2c (μ). A summary of the stability results as presented in Theorem 9.2 for strain 1 is obtained for strain 2 by replacing the parameter indices 1’s with 2’s and replacing f (R1 ) with g(R2 ). Functions f (R1 ) and g(R2 ) help determine the stability and coexistence regions for strains 1 and 2. In fact, changes in the regions of stability for a single and both strains can be illustrated by varying the coefficient of cross-immunity. For instance, from (9.32) we can compute the value of σ at which f ′ (R1 ) ≡ namely σ1∗ =  1+

δ2 μ+γ2

∂f (R1 , σ )   ∗ = 0, σ1 ∂R1 

(9.34)

1 1−

μ(μ+α1 ) (μ+γ1 )(μ+α1 )+α1 δ1

.

Hence, for all R1 > 1, f ′ (R1 ) > (< or =) 0,

f (R1 ) > (< or =) 1

if σ < (> or =) σ1∗ .

These properties can be easily checked by noticing from (9.32) that f (R1 ) =

R1 1 + σσ∗ (R1 − 1) 1

and

f ′ (R1 ) =  1+

1−

σ σ1∗

σ σ1∗ (R1

− 1)

2 .

Using the symmetry between two strains, we can show that similar properties hold for another threshold value σ2∗ (interchanging the subscripts 1 and 2 in the expression of σ1∗ ) and a function R1 = g(R2 ). The properties of f and g are illustrated in Fig. 9.11. The first two plots in Fig. 9.11 are for the special case when the two strains have identical parameter values (σ1∗ = σ2∗ = σ ∗ ), whereas the last two plots are for the case σ1∗ = σ2∗ . R2 < f (R1 ) is a necessary condition for the stability of strain 1 (either a stable boundary endemic equilibrium E1 or the equilibrium associated with strain 1 oscillations). Hence, E2 is unstable when R2 > f (R1 ). Similarly, E1 is unstable when R1 > g(R2 ). Hence, coexistence is expected when R2 > f (R1 ) and R1 > g(R2 ). Figure 9.12 depicts the stability of the equilibria and periodic solutions when (R1 , R2 ) lies in Regions I and III. It illustrates how the stabilities of equilibria and

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9 Models for Influenza

Fig. 9.11 Bifurcation diagrams in the (R1 , R2 ) plane for various combination of σ1 and σ2 values in relation to the threshold levels of σ1∗ and σ2∗ (cross-immunity). Regions I –I I I denote the existence and stability of E1 , E2 , and coexistence, respectively

Fig. 9.12 Depiction of the stability properties of equilibria and periodic solutions for parameter values in different regions. A solid circle represents a stable boundary (strain 1 only) equilibrium. A solid square represents an unstable boundary (strain 2 only) equilibrium. A star represents a stable interior (coexistence of both strains) equilibrium. A solid (dashed) closed orbit represents a stable (unstable) period solution

periodic orbits change their stability when the parameters α1 and α2 change their values crossing the critical points αic (i = 1, 2). The focus of this study is on the time evolution of influenza A in a non-fixed landscape driven by tight coevolutionary interactions (that is, interactions where the fate of the host and the parasite are intimately connected) between human hosts and competing strains. The process is mediated by intervention (behavioral changes) and cross-immunity. In other words, the nature of the invading landscape (susceptible host) changes dynamically from behavioral changes (isolation, short time scale) and past immunological experience (cross-immunity, long time scale). The “partial” herd immunity generated by past history of invasions on the host population can have a huge impact on the quantitative dynamics of the “flu” at the population level. The assumption that σ12 = σ21 = σ for i = j naturally results in a dynamic landscape that is not too different (in the oscillatory regime) than

9.8 Exercises

347

the one observed on single-strain models with isolation [25, 31]. That is, a lack of heterogeneity in cross-immunity results in a system “more or less” driven (in the oscillatory regime) by the process of isolation. The results show that multiple strain coexistence is highly likely for antigenically distinct (weak cross-immunity) strains and not for antigenically similar under symmetric cross-immunity (“competitive exclusion” principle [10]) . As the levels of cross-immunity weaken, the likelihood of sub-threshold coexistence increases. However, “full” understanding of the evolutionary implications that result from human host and influenza virus interactions may require the study of systems that incorporate additional mechanisms such as seasonality in transmission rates, age structure, individual differences in susceptibility or infectiousness, and the possibility of coinfections. Thacker [46] notes that the observed seasonality of influenza in temperate zones may be the key to observed patterns of recurrent epidemics. Superinfection may also be a mechanism worth consideration, even though studies in [46] show that it is only moderately possible for young individuals to become infected with two different strains in one “flu” season.

9.8 Exercises 1. Write the basic influenza model of Sect. 9.2 as an age of infection model and calculate its basic reproduction number. 2. Write the vaccination model of Sect. 9.2.1 as an age of infection model and calculate its reproduction number. [Warning: In order to do this, you will need to formulate a heterogeneous mixing age of infection model because vaccinated and unvaccinated individuals have different infectivities and susceptibilities.] In the analysis of seasonal and pandemic influenza in this chapter, we have used simple SI R models. However, influenza has a more complicated structure, with exposed periods and asymptomatics. The obvious way to include this more complicated structure would be to use age of infection models. 3. Formulate an age of infection model analogous to the seasonal epidemic model (9.13), and calculate its reproduction number and final size relation. 4. Formulate an age of infection model analogous to the pandemic influenza model (9.18), and calculate its reproduction number and final size relation. 5. Formulate an age of infection model analogous to the combined seasonal/pandemic influenza model (9.21) and calculate its reproduction number. 6. Consider the model (9.14) in Sect. 9.4. Choose parameters to make the basic reproduction number equal to 1.5, take σi = 1 − q i , where q = 0.75, and τi = 1. Simulate the model and determine the final sizes. 7. Use the final sizes obtained from Exercise 6 as initial sizes and repeat the simulation. Repeat this several times to see if the final sizes approach limits.

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9 Models for Influenza

8. Use the limiting final sizes obtained in Exercise 7 as initial sizes and simulate the model (9.18) of Sect. 9.5 with parameters giving a basic reproduction number of 2. Determine the final size of the pandemic. 9. Simulate the model (9.21) of Sect. 9.5 for various values of ρ between 0 and 1. For which values do seasonal and pandemic strains coexist and for which values does the pandemic strain replace the seasonal strain? (References: [3, 4, 8].)

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39. Nuno, M., Z. Feng, M. Martcheva, and C. Castillo-Chavez (2005) Dynamics of two-strain influenza with isolation and partial cross-immunity, SIAM J. Appl. Math. 65: 964–982. 40. Prosper, O., O. Saucedo, D. Thompson, G. Torres-Garcia, X. Wang, and C. Castillo-Chavez (2011) Control strategies for concurrent epidemics of seasonal and H1N1 influenza, Math. Biosc. Eng. 8: 147–177. 41. Public Health Agency of Canada, Vaccine Development Process, URL http://www.phac-aspc. gc.ca/influenza/pandemic-eng.php. 42. Qiu, Z. & Z. Feng (2010) Transmission dynamics of an influenza model with vaccination and antiviral treatment, Bull. Math. Biol. 72: 1–33. 43. Stillianakis, N.I., A.S. Perelson & F.G. Hayden (1998) Emergence of drug resistance during an influenza epidemic: insights from a mathematical model, J. Inf. Diseases, 177: 863–873. 44. Olinsky, R., A. Huppert & L. Stone (2008) Seasonal dynamics and threshold governing recurrent epidemics, J. Math. Biol. 56: 827–839. 45. Stone, L., R. Olinsky & A. Huppert (2007) Seasonal dynamics of recurrent epidemics, Nature 446: 533–536. 46. Thacker, S. B. (1986) The persistence of influenza A in human populations, Epidemiologic Reviews, 8: 129–142. 47. Towers, S., and Z. Feng (2009) Pandemic H1N1 influenza: Predicting the course of pandemic and assessing the efficacy of the planned vaccination programme in the United States, Eurosurveillance, 14(41): Article 2. 48. van den Driessche, P. & J. Watmough (2002) Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosc. 180:29–48. 49. Vivas-Barber, A., C. Castillo-Chavez, and E. Barany (2014) Dynamics of an “SAIQR” influenza model, Biomath, 3: 1–13. 50. Welliver, R., A.S. Monto, O. Carewicz, E. Schatteman, M. Hassman J. Hedrick, H.C. Jackson, L. Huson, P. Ward & J.S. Oxford (2001) Effectiveness of oseltamivir in preventing influenza in household contacts: a randomized controlled trial, JAMA 285: 748–754. 51. Wu, L. and Z. Feng (2000) Homoclinic bifurcation in an SIQR model for childhood disease, J. Differential Equations, 168: 150–167.

Chapter 10

Models for Ebola

Another important infectious disease is Ebola virus disease (EVD). Ebola hemorrhagic fever is a very infectious disease with a case fatality rate of more than 70%. It was first identified in 1976 in the Democratic Republic of Congo and there have been more than a dozen serious outbreaks since then. The most serious outbreak to date occurred in 2014 in Guinea, Liberia, and Sierra Leone and caused more than 10,000 deaths. Response to this epidemic included the development of vaccines to combat the disease, currently being used to combat the most recent outbreak in the Democratic Republic of Congo. A distinctive feature of the Ebola virus is much disease transmission occurs through contact with bodily fluids at funerals of Ebola victims. Many mathematical models have been used to study its disease transmission dynamics. Most of these studies have focused on estimating the basic and effective reproduction numbers of EVD, assessing the rate of growth of an epidemic outbreak, evaluating the effect of control measures on the spread of EVD, and conducting more theoretical investigations on how model assumptions may affect model outcomes (see, for example, [1, 3, 7, 8, 11, 18, 23, 25–27, 29, 31, 35, 36, 40]). Although some of these models have provided useful information and better understanding of EVD dynamics and evaluation of control programs, most of these models failed to provide reasonable projections for the 2014 outbreak in West Africa. It is important to examine the reasons for this, including the underlying assumptions made in these models. In this chapter, we describe several models for Ebola.

10.1 Estimation of Initial Growth and Reproduction Numbers Estimation of the basic and effective reproduction numbers for EVD has been conducted for both the 2014 outbreaks in West Africa and some of the outbreaks in the past (see, for example, [1, 11, 23, 25, 36]). In [11], a standard SEI R model © Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_10

351

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10 Models for Ebola

is used for estimating R0 and exploring the effect of timing of the intervention on the epidemic final size. Let S(t), E(t), I (t), and R(t) denote the number of susceptible, exposed, infective, and removed individuals at time t (the dot denotes time derivatives), and let C(t) denote the cumulative number of Ebola cases from the time of onset of symptoms. Assume that exposed individuals undergo an average incubation period (asymptomatic and uninfectious) of 1/k days before progressing to the infective class I . Infective individuals move to the R-class (death or recovered) at the per-capita rate γ . The model reads S ′ = −βS(t)I (t)/N E ′ = βS(t)I (t)/N − kE(t) I ′ = kE(t) − γ I (t) R ′ = γ I (t) C ′ = kE(t).

(10.1)

The model (10.1) predicts initial exponential growth of the number of infectives, but in fact the initial growth rate is less than exponential. One way to modify the model to allow this is to assume behavioral changes including education of hospital personnel and community members on the use of strict barrier nursing techniques (i.e., protective clothing and equipment, patient management), and the rapid burial or cremation of patients who die from the disease. It is assumed that the net effect is a reduced transmission rate β from β0 to β1 < β0 . To take into consideration that the impact of the intervention is not instantaneous, the transmission rate is assumed to decrease gradually from β0 to β1 according to β(t) =

%

β0 β1 + (β0 − β1 )e−q(t−τ )

t s = Pi (s) Probability of a living individual not being hospitalized s units of time since onset   for models I, II when i = 1, 2, respectively. That is, P TLi > s = Li (s) Probability of surviving the disease s units of time since onset for model I. That is,   P TM1 > s = M1 (s) Probability of not having recovered s time units after being hospitalized for model III Mean duration from onset to recovery (absent intervention or death) Mean duration from onset to hospitalization (given hospitalized and not dead) Mean duration between onset and death (absent intervention or recovery) Mean duration in the I compartment (hospitalization and death included) Mean duration in the H compartment (death included) Mean duration from hospitalization to recovery Mean duration from hospitalization to death = 1/E[TP ] = 1/E[TL ] = 1/E[TM ] = 1/DH R , per-capita rate of transition from H to R if the transition is exponential = 1/DH D , per-capita rate of transition from H to D if the transition is exponential Proportion hospitalized (dependent on control effort) Probability of death (with or without hospitalization) = 1/E[XI ], per-capita rate of exiting I if XI is exponential

If the transitions IR, IH, and ID are assumed to be independent in the Legrand model and the waiting times are all exponentially distributed with mean durations 1/γI R , 1/γI H , and 1/γI D , respectively, then the mean overall time spent in the I compartment is   E min {TP , TL , TM } =



0

∞

P (t)L(t)M(t)dt =

1 γ I R + γI H + γI D

.

10.4 *Models with Various Assumptions on Stage Transition Times

367

Thus, the overall rate of exiting I is γI R + γI H + γI D , which is not a weighted average given by Δ in (10.11) for the Legrand model. This implies that the Legrand model has made different assumptions on these transitions. In the formulation of the integro-differential equations models, we will adopt probabilistic terminology to facilitate the interpretation of these models, and focus on the following three scenarios: (I) Assume that the three transitions IR, IH, and ID are independent and the waiting times are described by the survival functions P1 (s), L1 (s), and M1 (s), respectively, where s represents the time-since-onset. It is also assumed that hospitalization does not affect the time from onset to recovery or death. (II) The two transitions IR and ID are combined and described by a single survival function P2 (s), with a fraction 1 − f of the exiting individuals recovering (and the fraction f dying). The transition IH is independent of IR and ID and the waiting time is described by the survival function L2 (s). Similar to model I, it is also assumed that hospitalization does not affect the time from onset to recovery or probability of death. (III) All three transitions (IH, IR, and ID) are combined and described by a single survival function P3 (s), with a fraction p of the exiting individuals being hospitalized and a fraction 1 − f (respectively, f ) of the non-hospitalized individuals recovering (respectively, dying). The two transitions HR and HD are combined and the waiting time is described by a single survival function Q3 (s) with a fraction 1 − f (or f ) of the exiting individuals recovering (or dying). P3 and Q3 are assumed to be independent. Unlike models I and II, in which the time from onset to hospitalization is tracked due to the independent stage distributions, in model III a constraint must be imposed so that the time between onset and hospitalization plus the time between hospitalization and recovery (or death) equals the time between onset and recovery (or death). To focus on the general waiting time for the infective stage and its influence on model formulation when hospitalization is considered, assume simpler distributions for other stages including the latent stage and the duration between death and burial. That is, the E and D stages are assumed to have exponential distributions with constant rates α and γf . As the models are derived under arbitrary distributions for the waiting times of key disease stages, they consist of systems of integrodifferential equations. It shows that these systems reduce to ODE systems when the arbitrary stage distributions are replaced by gamma or exponential distributions. Detailed derivations of the systems of integral equations are provided in Feng et al. (2016).

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The model I, which corresponds to the scenario (I) described above, assumes independent IR, IH, and ID processes. Let TP1 , TL1 , and TM1 denote random variables for the independent waiting times of IR, IH, and ID, respectively, that are described by the following survival functions: • P1 (s): Probability that a living individual remains infectious s units of time since onset (governing both IR and HR). • L1 (s): Probability of a living individual not being hospitalized s units of time since onset (governing the IH transition). • M1 (s): Probability of surviving the disease s units of time since onset (governing both ID and HD). Figure 10.10 depicts the transitions between epidemiological classes for the model under scenario (I). All variables and parameters have the same meanings as before unless otherwise stated. The diagram in (a) depicts transitions between compartments when stage durations for the IR, IH, and ID transitions are arbitrarily described by the survival functions P1 (t), L1 (t), and M1 (t). The dotted rectangle around the I and H compartments indicates that individuals in these two compartments are being tracked for their time-since-onset using the same survival

(a) Arbitrary distributions

(b) Gamma distributions

Fig. 10.10 Transition diagram for Model I when TP1 , TL1 , and TM1 are arbitrary (a), or gamma/exponential (b). The corresponding survival functions are P1 (t) (red), L1 (t) (green), and M1 (t), respectively

10.4 *Models with Various Assumptions on Stage Transition Times

369

function P1 (t), i.e., the time elapsed in I before entering H is taken into account when determining the time between entering H and recovery. The diagram in (b) illustrates the effect of the “linear chain trick” when P1 (t) follows a gamma distribution, and L1 (t) and M1 (t) follow exponential survival functions. The model with the general stage distributions P1 , L1 , and M1 consists of the following system of integro-differential equations: dE = λ(t)S − αE, dt

dS = −λ(t)S, dt I (t) =



H (t) =



D(t) = R(t) =

t

αE(s)P1 (t − s)L1 (t − s)M1 (t − s)ds + I (0)P1 (t)L1 (t)M1 (t),

0 t

0

  αE(s)P1 (t − s)M1 (t − s) 1 − L1 (t − s) ds

  +I (0)P1 (t)M1 (t) 1 − L1 (t) ,

 t  0

 t  0

τ

0 τ 0

(10.16) 

αE(s)P1 (τ − s)gM1 (τ − s)ds + I (0)P1 (τ )gM1 (τ ) e−γf (t−τ ) dτ,  αE(s)gp1 (τ − s)M1 (τ − s)ds + I (0)gp1 (τ )M1 (τ ) dτ,

where λ(t) is given by λ(t) =

βI I + βH H + βD D , N

(10.17)

The initial condition is (S(0), E(0), I (0), H (0), D(0), R(0)) = (S0 , E0 , I0 , 0, 0, 0), where S0 and E0 are positive constants. Notice that in system (10.16), the probability distributions for TP1 , TL1 , and TM1 are arbitrary. Consider the case when TP1 follows a gamma distribution with shape and rate parameters (n, nγ1 ) (where n ≥ 1 is an integer), and TL1 and TM1 follow exponential distributions with parameters χ1 and μ, respectively (which are gamma distributions with shape parameter 1). That is, P1 (t) = Gnnγ1 (t) =

n (nγ1 t)j −1 e−nγ1 t j =1

L1 (t) = G1χ1 (t) = e−χ1 t , M1 (t) = G1μ (t) = e−μt .

(j − 1)!

, (10.18)

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10 Models for Ebola

Then, it is shown in [18] that (10.16) is equivalent to the following system of ODEs: n n  dS 1  Ij + βH Hj + βD D , = − S βI dt N j =1 j =1 n n  1  dE = S βI Ij + βH Hj + βD D − αE, dt N j =1 j =1 dI1 = αE − (nγ1 + χ1 + μ)I1 , dt dIj = nγ1 Ij −1 − (nγ1 + χ1 + μ)Ij , for j = 2, . . . n, dt dH1 = χ1 I1 − (nγ1 + μ)H1 , dt dHj = χ1 Ij + nγ1 Hj −1 − (nγ1 + μ)Hj , for j = 2, . . . n, dt n n dD =μ Hj − γf D, Ij + μ dt j =1 j =1 dR = nγ1 In + nγ1 Hn . dt

(10.19)

In the special case when n = 1 (i.e., P1 is also an exponential distribution), the model (10.19) simplifies to 1 dS = − S(βI I + βH H + βD D), dt N 1 dE = S(βI I + βH H + βD D) − αE, dt N dI = αE − (γ1 + χ1 + μ)I, dt dH = χ1 I − (γ1 + μ)H, dt dD = μI + μH − γf D, dt dR = γ1 I + γ1 H. dt

(10.20)

10.4 *Models with Various Assumptions on Stage Transition Times

371

Note that in model (10.20) the per-capita transition rates from I to R and from H to R are both equal to γ1 , from which we have γI R = ωH R . This means that the constraints in (10.15) for model (10.12) cannot be satisfied. Thus, model I cannot be equivalent to the Legrand model. This suggests that Legrand et al. did not assume that the three transitions of IR, IH, and ID were described by independent exponential distributions and that the overall waiting time in the I compartment was the minimum of these three exponential waiting times. When P1 , L1 , and M1 are exponential with the respective parameters γ1 , χ1 , and μ, because of the assumption in model I that the three transitions IR, IH, and ID are independent, the overall waiting time in the I compartment is also exponential with the rate constant γ1 + χ1 + μ. From the definition of these parameters, we can link them to the general parameters (i.e., independent of model assumptions) in Table 10.2. There might be multiple ways of making the connections. One example is the following: For example, 1 1 = = E[TP1 ], γ1 γI R

f =

μ , γ1 + μ

p=

χ1 , γ 1 + χ1 + μ

(10.21)

where γI R , f , and p are parameters that are independent of models. From the relations in (10.21) we can express the rates γ1 , χ1 , and μ in terms of only γI R , p, and f : γ1 = γI R ,

χ1 = γ I R

p , (1 − f )(1 − p)

μ = γI R

f . 1−f

(10.22)

Reproduction Numbers for Models (10.16), (10.19), and (10.20) Based on the biological meaning of RC , we obtain the following expression for general RC1 for model (10.16) under general stage distributions: general

RC1

3

2 = βI E min TP1 , TL1 , TM1 3 3

2 

2 + βD γ1f pM , +βH E min TP1 , TM1 − E min TP1 , TL1 , TM1

(10.23)

where

2 3 E min TP1 , TL1 , TM1 =



∞ 0

P1 (t)L1 (t)M1 (t)dt

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10 Models for Ebola

represents the average time spent in the I compartment, and 3

2 E min TP1 , TM1 =



∞

P1 (t)M1 (t)dt

0

represents the average total time spent in the I and H compartments. For the model (10.19) with gamma distribution, Gamma = RC1

  n  nγ1 βI 1− χ1 + μ nγ1 + χ1 + μ   n  n  χ1 nγ1 μ βH nγ1 + − + μ χ1 + μ χ1 + μ nγ1 + χ1 + μ nγ1 + μ n    nγ1 βD . 1− + γf nγ1 + μ (10.24)

For the model (10.20) with exponential distribution, Exp

RC1 =

βI βH χ1 βD μ + + γ 1 + χ1 + μ γ 1 + μ γ 1 + χ1 + μ γf γ1 + μ

βH p βD f βI + + = . γ 1 + χ1 + μ γ 1 + μ γf

(10.25)

Next, consider different assumptions on the transition processes. In model I, the three transitions IR, IH, and ID are assumed to be independent, in which case the three transitions “compete” for individuals in the I class. Another scenario is to consider only two independent transitions, one being hospitalization and the other combining recovery and death for those who are not hospitalized. In this case, we have two survival probability functions: • P2 (s): Probability of still being infectious and alive s time units after onset (governing all four transitions IR, ID, HR, and HD). • L2 (s): Probability of not being hospitalized s time units after onset (governing the IH transition). Assume that, for those who exit I without being hospitalized, a fixed fraction 1 − f (or f ) will recover (or die) an assumption of Legrand model. Assume also that individuals in I and H classes have the same probability of death (f ), also as assumed in the Legrand model. The transition diagram is depicted in Fig. 10.11a.

10.4 *Models with Various Assumptions on Stage Transition Times

373

(a) Arbitrary distributions

(b) Gamma distributions

Fig. 10.11 A transition diagram for model II when TP2 and TL2 are arbitrary distributions (a) and when they are gamma or exponential (b). In (a), the recovery/death (red) and hospitalization (green) transitions are governed by the survival functions P2 and L2 . In (b), the recovery/death (red) and hospitalization (green) transitions are indicated by the same colors as in (a)

Model II has the form dS(t) = −λ(t)S(t), dt dE(t) = λ(t)S(t) − αE(t), dt  t I (t) = αE(s)P2 (t − s)L2 (t − s)ds + I (0)P2 (t)L2 (t), 0  t     H (t) = αE(s)P2 (t − s) 1 − L2 (t − s) + I (0)P2 (t) 1 − L2 (t) , 0   t  τ D(t) = f αE(s)gP2 (τ − s)ds + I (0)gP2 (τ ) e−γf (t−τ ) dτ, 0 0   t  τ R(t) = (1 − f ) αE(s)gP2 (τ − s)ds + I (0)gP2 (τ ) dτ. 0

0

(10.26)

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The function λ(t) is the same as in model I and given in (10.17). We can reduce the integral equations in (10.26) to the ODEs given below: n n  dS 1  Ij + βH Hj + βD D , = − S βI dt N j =1 j =1 n n  dE 1  = S βI Ij + βH Hj + βD D − αE, dt N j =1 j =1 dI1 = αE − (nγ2 + χ2 )I1 , dt dIj = nγ2 Ij −1 (t) − (nγ2 + χ2 )Ij , for j = 2, . . . n, dt dH1 = χ2 I1 − nγ2 H1 , dt dHj = χ2 Ij + nγ2 Hj −1 − nγ2 Hj , for j = 2, . . . n, dt dD = f nγ2 In + f nγ2 Hn − γf D, dt dR = (1 − f )nγ2 In + (1 − f )nγ2 Hn . dt

(10.27)

The reproduction numbers RC2 for model II also have different forms than those for model I. In the case of general distributions, general

RC2

3

2 3  

2 = βI E min TP2 , TL2 + βH E TP2 − E min TP2 , TL2 +βD γ1f [f (1 − pH 2 ) + fpH 2 ] .

(10.28)

When P2 is a gamma distribution,   n  1 nγ2 1− χ2 nγ2 + χ2    n  1 nγ2 f 1 +βH − 1− + βD . γ2 χ2 nγ2 + χ2 γf

Gamma = β RC2 I

(10.29)

10.4 *Models with Various Assumptions on Stage Transition Times

375

When P2 is exponential, the model (10.27) becomes dS 1 = − S(βI I + βH H + βD D), dt N dE 1 = S(βI I + βH H + βD D) − αE, dt N dI = αE − (γ2 + χ2 )I, dt dH = χ2 I − γ2 H, dt dD = f γ2 I + f γ2 H (t) − γf D(t), dt dR = (1 − f )γ2 I + (1 − f )γ2 H. dt

(10.30)

The formula for RC2 in (10.28) simplifies to Exp

RC2 =

pβH fβD βI + + . γ 2 + χ2 γ2 γf

(10.31)

As in model I, there can be multiple choices for linking the parameter γ2 to the common parameters. For example, 1 1 1 = (1 − f ) +f , γ2 γI R γI D where 1/γI D denotes the average time from onset to death. Also, p = χ2 /(γ2 +χ2 ). Thus, γ2 =

1 , (1 − f )/γI R + f/γI D

χ2 =

γ2 p . 1−p

(10.32)

Another set of possible assumptions that are different from models I and II is to consider two independent distributions for the waiting times in I and H compartments, denoted by TP3 and TQ3 , with survival functions • P3 (s): Probability of remaining in the I class s units of time since onset (governing the transitions IR, IH, and ID). • Q3 (s): Probability remaining in the H class s units of time after being hospitalized (governing the HR and HD transitions). Let p3 (t) = −P3′ (t) and q3 (t) = −Q′3 (t) denote the probability density functions. P3 describes the waiting time for the combined transitions, IR, IH, and ID. Assume that among the individuals exiting the I class, fractions of p, (1−p)f , (1−p)(1−f ) will be hospitalized, non-hospitalized and dead, and non-hospitalized and recovered, respectively (0 ≤ p, f < 1). Q3 describes the waiting time for the combined two

376

10 Models for Ebola (a) Arbitrary distributions

(b) Gamma distributions

Fig. 10.12 A transition diagram for model III when TP3 and TQ3 are arbitrary distributions (a) and when they are gamma or exponential (b)

transitions, HR and HD, and we assume that fractions 1−f and f of the hospitalized individuals recover or die, respectively. A transition diagram is shown in Fig. 10.12. In this case, model III consists of the following system of integro-differential equations: S ′ (t) = −λ(t)S(t), E ′ (t) = λ(t)S(t) − αE(t),  t I (t) = αE(s)P3 (t − s)ds + I (0)P3 (t),  0 t  s αE(τ )p3 (s − τ )dτ + I (0)p3 (s) Q3 (t − s)ds p H (t) = 0  0  t ′ D (t) = (1 − p)f αE(s)p3 (t − s)ds + I (0)p3 (t)   t  s 0 p αE(τ )p3 (s − τ )dτ + pI (0)p3 (s) q3 (t − s)ds − γf D(t), +f 0 0  t  ′ R (t) = (1 − p)(1 − f ) αE(s)p3 (t − s))ds + I (0)p3 (t)   t  s 0 p αE(τ )p3 (s − τ )dτ + pI (0)p3 (s) q3 (t − s)ds. +(1 − f ) 0

0

(10.33)

10.4 *Models with Various Assumptions on Stage Transition Times

377

Note that it is easier in this case to write equations for D ′ and H ′ than for D and H . Assume that TP3 and TQ3 follow gamma distributions with shape parameters n and m respectively, i.e., the survival functions are given by P3 (t) = Gnnγ3 (t) = Q3 (t) =

Gm mω3 (t)

n [nγ3 (t − s)] j −1 e−nγ3 (t−s) , (j − 1)! j =1 m

=

j =1

[mω3 (t − s)] j −1 e−mω3 (t−s) . (j − 1)!

(10.34)

Then, the system (10.33) reduces to the following system of ODEs: n m  dS 1  = − S βI Ij + βH Hj + βD D , dt N j =1 j =1 n m  dE 1  = S βI Ij + βH Hj + βD D − αE, dt N j =1 j =1 dI1 = αE − nγ3 I1 , dt dIk = nγ3 Ik−1 − nγ3 Ik , k = 2, . . . , n (10.35) dt dH1 = pnγ3 In − mω3 H1 , dt dHk = mω3 Hk−1 − mω3 Hk , k = 2, . . . , m dt dD = (1 − p)f nγ3 In + f mω3 Hn − γf D, dt dR = (1 − p)(1 − f )nγ3 In + (1 − f )mω3 Hm . dt A transition diagram under the gamma distributions for TP3 and TQ3 , for the ODE model (10.35) is shown in Fig. 10.12b. We observe a major difference between this figure and Fig. 10.10b or Fig. 10.11b in the recovery rates from In and Hm , which have different values here. A similar difference exists in the transition rates from Ij to Ij +1 (j = 1, · · · , n − 1) and from Hj to Hj +1 (j = 1, · · · , m − 1).

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10 Models for Ebola

In the special case when n = m = 1 (i.e., P3 and Q3 are exponential), the ODE model (10.35) simplifies to 1 dS = − S(βI I + βH H + βD D), dt N dE 1 = S(βI I + βH H + βD D) − αE, dt N dI = αE − γ3 I, dt dH = pγ3 I − ω3 H, dt dD = (1 − p)f γ3 I + f ω3 H − γf D, dt dR = (1 − p)(1 − f )γ3 I + (1 − f )ω3 H. dt

(10.36)

Notice that, if we ignore the last term in the R equation in the equivalent Legrand model (10.12) (this term indicates that the R class includes those buried), the model (10.36) is identical to the model (10.12) when the subscript “3” is dropped; that is, when γ3 and ω3 are defined as follows: 1 p (1 − p)f (1 − p)(1 − f ) = + + , γ3 γI H γI D γI R 1 f 1−f = + , ω3 ωH D ωH R

(10.37)

together with the constraints 1 γI R

=

1 γI H

+

1 ωH R

,

1 γI D

=

1 γI H

+

1 ωH D

.

(10.38)

The reproduction number RC3 for model III can be derived using the same approach as for models I and II. Because the derivations are similar, we omit the details, and present the formula only for the special case when P3 and Q3 are both exponential, i.e., P3 = G1γ3 and Q3 = G1ω3 (see (10.34)). In this case, the formula for RC3 for the ODE model (10.35) is independent of m and n, and is given by Exp

RC3 =

βH βD βI +p +f , γ3 ω3 γf

(10.39)

where γ3 and ω3 are given in (10.37). Note that the main difference between models I, II, and III lies in the assumptions on the underlying biological processes, particularly the sojourn distributions for

10.4 *Models with Various Assumptions on Stage Transition Times

379

various stage transitions, which are described by functions L(t), P (t), M(t), and Q(t). The fact that the Legrand model can only be obtained from model III, not from models I and II, identifies the specific assumptions made in the Legrand model in terms of these sojourn distributions. For example, our analyses suggest the following assumptions made in the Legrand model: (a) The overall sojourn in the I stage is assumed to be exponentially distributed with the average duration 1/γ , which is further assumed to be the specific weighted average of 1/γI R , 1/γI H , and 1/γI D as given in (10.13), where 1/γI R , 1/γI H , and 1/γI D are the respective average stage duration of the IR, IH, and ID transitions. However, from model I we see that if the IR, IH, and ID transitions follow independent exponential distributions with parameters γI R , γI H , and γI D , then the overall sojourn in the I stage is exponentially distributed with the parameter γ = γI R + γI H + γI D with the average duration 1 1 = , γ γI R + γI H + γI D which differs from (10.13). (b) The distributions for the I and H stages are independent (see P3 (t) and Q3 (t)). This implies that the average time spent in the H stage before recovery or death (1/ω3 ) does not depend on the average time spent in the I stage before recovery or being hospitalized or dying (1/γ3 ). Under this assumption, the time spent in H before recovery (1/γH R ) does not take into consideration the time spent in I before being hospitalized (1/γI H ) . Because of this independence, the model needs to impose a constraint to link these two durations (see (10.10)). Difference in Evaluations by Models I, II, and III Among the three ODE models (10.20), (10.30), and (10.36), which are reduced from the models I, II, and III with general distributions, the only model that can match the Legrand model is (10.36), for which the assumptions include: (i) the waiting times of the three transitions IR, IH, and ID are not independent and (ii) the overall waiting time in the I compartment is a weighted average of the mean durations (1/γI R , 1/γI H , and 1/γI D ) for the three transitions with weights determined by the probabilities of hospitalization p and death f , as described in (10.14). By examining the ODE model (10.20), we found that, if the waiting times of the three transitions IR, IH, and ID are independent and exponentially distributed (with parameters γ1 , χ1 , and μ), then the overall waiting time in I should be an exponential distribution with the parameter γ1 + χ1 + μ. That is, the average overall waiting time should be 1/(γ1 + χ1 + μ), not a weighted average such as the ones in (10.14). Formulas for the control reproduction numbers RCi (i = 1, 2, 3) for the three general models provide a means of examining the influence of assumptions on model outcomes. For example, consider the three control reproduction numbers RCi (i = 1, 2, 3), which are given in (10.25), (10.31), and (10.39) corresponding to

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Fig. 10.13 Plots of the basic reproduction numbers (a) and control reproduction numbers (b) for the three models. In (a), R0 is plotted as a function of f for model I (thin solid), model II (dashed), and model III (thick solid). The parameter values are chosen such that all three R0i have the same value 1.8 at f = 0.7 (note that p = 0). In (b), RCi is plotted as a function of p and f for models I—III. Other parameter values are given in the text

the three ODE models (10.20), (10.30), and (10.36), respectively. In the absence of hospitalization (i.e., p = 0), these RCi reduce to the corresponding basic reproduction numbers R0i (i = 1, 2, 3). Figure 10.13 illustrates the difference between the basic and control reproduction numbers of the three models for a given set of parameter values, mostly based on the Ebola outbreak in West Africa in 2014. We fix all parameters except βi (i = I, H, D) and f . Then, for a fixed value of f0 = 0.7, we estimate βi (i = I, H, D) from a given value of R0 (assumed to be the same for all three models). If we further assume that βH = βD = 0.3βI , then we can get a unique value for βI for fixed R0 . Once we have the value of βi , we have three functions of f , R0i (f ) (i = 1, 2, 3). For Fig. 10.13a, we used the common value of R0 (f0 ) = 1.8. The three curves are for model I (the thin solid curve), model II (the dashed curve), and model III ( the think curve). For the selected set of parameter values, the R0 curves for models II and III overlap. The decreasing property of these curves represents the fact that higher disease mortality decreases R0i (f ), which is expected because the assumption that βD < βI . An interesting observation is that the dependence of the basic reproduction number on disease death f is more dramatic in model I than models II and III, particularly for smaller f values. For smaller values of f , model I tends to generate the highest R0 , while for larger f values, model III provides higher R0 . Other parameter values used are (time in days): 1/γI R = 18, 1/γf = 2, 1/α = 9. Parameters such as μ, χi (i = 1, 2) are calculated based on their relationships with the common parameters. For model III, additional parameter values include 1/γI H = 7, 1/γI D = 8, which can be used to determine γ3 and ω3 from (10.37). When control is considered (p > 0), the dependence of RCi on p and f is illustrated in Fig. 10.13b. We observe that, for the given set of parameter values, model I (the darker surface with mesh) generates higher RC values than models

10.4 *Models with Various Assumptions on Stage Transition Times

381

II (the lighter surface) and III (the darker surface with no mesh) for smaller p and f , while model III provides the higher values for larger values of p and/or f . The differences in R0 and RC between the three models indicate that model predictions about and evaluations of the effectiveness of control measures could be very different as well. Figure 10.14 shows numerical simulation results of the three ODE models (10.20), (10.30), and (10.36), which are reduced from the models I, II, and III, and presented in columns 1, 2, and 3, respectively. The A, B, and C panels correspond to three sets of (p, f ) values: (p, f ) = (0, 0.7) (top panel), (p, f ) = (0.3, 0.5) (middle panel), and (p, f ) = (0.4, 0.7) (bottom panel). The top panel (A1–A3) is for the case of no hospitalization (p = 0). We observe that models II and III generate similar epidemic curves (fractions of infected individuals (E + I + H )/N), including peak sizes, times to peak, duration of epidemic (which lead to similar epidemic final sizes). Model I shows a higher peak size and an earlier time to peak. The middle panel (B1–B3) is for the case when the hospitalization is p = 0.3, and we observe that model I has the highest peak size while model II has the lowest. This is in agreement with the relative magnitudes of the control

Fig. 10.14 Numerical simulations of the three ODE models (10.20), (10.30), and (10.36), which are reduced from the models I, II, and III, respectively. The fractions of infected individuals (E + I +H )/N and death (D/N ) are plotted over a time period of 1000 days. Three sets of (p, f ) values are used: (p, f ) = (0, 0.7) (top row), (p, f ) = (0.3, 0.5) (middle row), and (p, f ) = (0.4, 0.7) (bottom row). Parameter values are the same as in Fig. 10.13

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reproduction numbers RCi , as the point (p, f ) = (0.3, 0.5) lies in the region where RC1 > RC3 > RC2 (see Fig. 10.13). For the bottom panel (C1–C3), because the point (p, f ) = (0.4, 0.7) lies in the region where RC3 > RC1 > RC2 , we observe that model III generates the highest peak size while model II again has the lowest.

10.5 Slower than Exponential Growth It has been standard practice in analyzing disease outbreaks to formulate a dynamical system as a deterministic compartmental model, then to use observed early outbreak data to fit parameters to the model, and finally to analyze the dynamical system to predict the course of the disease outbreak and to compare the effects of different management strategies. In general, such models predict an initial stochastic stage (while the number of infectious individuals is small), followed by a period of exponential growth. Measurement of this early exponential growth rate is an essential step in estimating contact rate parameters for the model. A thorough description of the analysis of compartmental models may be found in [21]. However, instances have been noted where the growth rate of an epidemic is clearly slower or faster than exponential. For example, [13], the 2013–2015 epidemic in West Africa has been viewed as a composition of locally asynchronous outbreaks at local levels displaying sub-exponential growth patterns during several generations. Specifically, if I (t) is the number of infectious individuals at time t, a graph of log I (t) against t is a straight line if the growth rate is exponential, and for some disease outbreaks this has not been true. One of the earliest examples [16] concerns the growth of HIV/AIDS in the USA, and a possible explanation might be the mixture of short-term and long-term contacts. This could be a factor in other diseases where there are repeated contacts in family groups and less frequent contacts outside the home. During the 2013–2015 Ebola epidemic in West Africa, in the country of Liberia, which has a total population size of 4,300,000, there were 10,678 suspected, probable, and confirmed cases of disease as of September 3, 2015 [32]. An SIR model for the whole country would have predicted more than 1,500,000 cases. This discrepancy cannot be explained by the assumption of control measures. In order to make plausible predictions of the effects of large-scale epidemics, it is necessary to use a phenomenological model based on observations in the early stages of the epidemic rather than to try to fit a mechanistic model. Some phenomenological models that have been used to good effect have been the generalized Richards model, the “generalized growth model,” and the IDEA model.

10.5 Slower than Exponential Growth

383

10.5.1 The Generalized Richards Model Perhaps the first attempt to fit early epidemic data is the Richards model [34]. This is a modification of the logistic population growth model, described in [22]   a  I . I (t) = rI 1 − K ′

In this model, I represents the cumulative number of infected individuals at time t, K is the carrying capacity or total case number of the outbreak, r is the per-capita growth rate of the infected population, and a is an exponent of deviation from the standard logistic model. The basic premise of the model is that the incidence curve has a single turning point tm . The analytic solution of the model is K I (t) =  1/a . 1 + e−r(t−tm )

10.5.2 The Generalized Growth Model It has been pointed out [12–15, 39] that a so-called general growth model of the form C ′ (t) = rC(t)p , where C(t) is the number of disease cases occurring up to time t, and p, 0 ≤ p ≤ 1, is a “deceleration of growth” parameter, has exponential solutions if p = 1 but solutions with polynomial growth if 0 < p < 1. This is not and does not claim to be a mechanistic epidemic model, but it has proved to be remarkably successful for fitting epidemic growth and predicting the course of an epidemic. For example, it has provided much better estimates of epidemic final size than the exponential growth assumption for the Ebola epidemic of 2014. However, it assumes a sustained increase in the number of disease cases and cannot capture the later decline in the number of new infections. Such phenomenological models are particularly likely to be suitable in situations where it is difficult to construct a mechanistic approach because of multiple transmission routes, interactions of spatial influences, or other aspects of uncertainty.

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10.5.3 The IDEA Model Another direction that would be well worth further exploration would be contact rates decreasing in time because of individual behavioral changes in response to a disease outbreak. A contact rate which is a decreasing function of time can certainly lead to early epidemic growth slower than exponential. A step in this direction has been initiated in a discrete model [19] that has been applied to an Ebola model in [20]. The IDEA model [19, 20, 37] is a discrete model that assumes damping of recruitment of new infections because of spontaneous or planned behavioral changes. It is assumed that the time interval in the model is equal to the duration of infection, so that the number of infective individuals at each stage is equal to the number of new infections. The resulting model is 

R0 It = (1 + d)t

t

.

Here, d is a discount factor describing the recruitment damping. A variety of epidemiological situations in which slower than exponential epidemic growth might be possible have been described. Ultimately, the challenge for epidemiological modeling would be to determine which of these situations allow slower than exponential growth by deriving and analyzing mechanistic models to describe each of these situations. This is an important new direction for epidemic modeling. Some suggestions include metapopulation models with spatial structure including cross-coupling and mobility, clustering in spatial structure, dynamic contacts, agent-based models with differences in infectivity and susceptibility of individuals, and reactive behavioral changes early in a disease outbreak. It may well turn out that slower than exponential growth may be ruled out in some cases but is possible in others. For example, heterogeneity of mixing in a single location can be modeled by an autonomous dynamical system and the linearization theory of dynamical systems at an equilibrium shows that early epidemic growth for such a system is always exponential. On the other hand, metapopulation models may well allow many varieties of behaviors. An important broader question is the matter of what information influences the behavior of people during a disease outbreak, and how to include this in a model. A recent book [28] describes some studies in this direction.

10.5.4 Models with Decreasing Contact Rates There is anecdotal evidence that in an outbreak of a disease considered very serious there is early action to decrease the risk of being infected by decreasing contact rates. This early action may begin even before government efforts to combat the disease.

10.6 Project: Slower than Exponential Growth

385

Since Ebola is a very serious disease, with case fatality rates of 70% or more, it is reasonable to assume a decreasing contact rate. This has been suggested in [1, 2]. In [1], an SI R model is assumed with a contact rate that decreases exponentially in time, and observations are used to estimate the rate of decrease. In [6], such a model is used to estimate the final size of an Ebola epidemic over a country using early growth rate data and this yields early final size estimates that are much closer to the eventual outbreak data than estimates assuming a constant contact rate. For example, for the 2014-5 Ebola outbreak in Guinea, a country with a total population size of 10,589,000 the assumption of a constant contact rate corresponding to R0 = 1.5 led to an estimate of more than 9,000,000 cases over the whole country, while a decreasing contact rate assumption led to an estimate of about 27,000 disease cases. The actual number of Ebola cases in Guinea in this outbreak was fewer than 4,000. The models described here are for an entire country and are not intended to replace more detailed models needed for disease management describing the progress of the disease in individual villages.

10.6 Project: Slower than Exponential Growth We have suggested in Sect. 10.5.4 that slower than exponential growth of the number of infectious individuals may be explained by assuming a time-dependent decrease in the contact rate. The question of what modeling assumptions can lead to slower than exponential growth is a complex one [13]. Another possible explanation might be a decrease in contact rate depending on the current state of the system. If the model system remains autonomous, growth will remain exponential so long as the contact rate is a differentiable function of I so that the theory of linearization at a disease-free equilibrium remains valid. However, if the rate of new infections is a NS f (I ) with f (I ) not differentiable at I = 0, different behavior is possible. Consider an SI R model in which the equation for I is I′ = β

S α I − γ I, N

with 0 < α < 1. Initially, so long as S ≈ N, we approximate this equation by I ′ = βI α − γ I.

(10.40)

Question 7 Show that the basic reproduction number is R0 =

β . γ

Question 8 Make the change of dependent variable u = log I and derive the differential equation satisfied by u.

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Exponential growth of I corresponds to linear growth of u. It is clear that if α = 1, u does grow linearly. Question 9 Determine whether the rate of growth of u is less than linear for α < 1, either by explicit solution of the equation for u or by simulations with β > γ (so that R0 > 1) and a range of values of α.

10.7 Project: Movement Restrictions as a Control Strategy Cordons Sanitaire or “sanitary barriers” are designed to prevent the movement, in and out, of people and goods from particular areas. The effectiveness of the use of cordons sanitaire has been controversial. This policy was last implemented nearly 100 years ago [9]. In desperate attempts to control disease, Ebola-stricken countries enforced public health officials decided to use this medieval control strategy in the EVD hot-zone, that is, the region of confluence of Guinea, Liberia, and Sierra Leone [30]. In this project, a framework that allows, in the simplest possible setting, the possibility of assessing the potential impact of the use of a Cordon Sanitaire during a disease outbreak is introduced. We consider an SI R epidemic model in two patches, one of which has a significantly larger contact rate, with short-term travel between the two patches. The total population resident in each patch is constant. We follow a Lagrangian perspective, that is, we keep track of each individual’s place of residence at all times [4, 17]. This is in contrast to an Eulerian perspective, which describes migration between patches. Thus we consider two patches, with total resident population sizes N1 and N2 , respectively, each population being divided into susceptibles, infectives, and removed members. Si and Ii denote the number of susceptibles and infectives, respectively, who are residents in Patch i, regardless of the patch in which they are present. Residents of Patch i spend a fraction pij of their time in Patch j , with p11 + p12 = 1,

p21 + p22 = 1.

The contact rate in Patch i is βi , and we assume β1 > β2 . Each of the p11 S1 susceptibles from Group 1 present in Patch 1 can be infected by infectives from Group 1 and from Group 2 present in Patch 1. Similarly, each of the p12 S1 susceptibles present in Patch 2 can be infected by infectives from Group 1 and from Group 2 present in Patch 2.

10.8 Project: Effect of Early Detection

387

Question 1 Show that the model equations are     I1 I1 I2 I2 − pi2 Si p21 + p12 + p22 Si′ = −pi1 Si p11 N1 N2 N1 N2     I I I I 1 2 2 1 + pi2 Si p21 − γ Ii , + p21 + p22 Ii′ = pi1 Si p11 N1 N2 N1 N2

i = 1, 2.

Question 2 Use the next generation matrix [38] to calculate the basic reproduction number. Question 3 Use the approach described in Chaps. 4 and 5 to determine the final size relations. Imposition of a Cordon Sanitaire amounts to replacing the fractions of normal travel between groups by p11 = p22 = 1,

p12 = p21 = 0.

Question 4 Compare the reproduction numbers and final sizes with and without a Cordon Sanitaire, using numerical simulations with various parameter value choices. The approach suggested here can be used with more detailed models for a specific disease, such as Ebola, in which transmission from one village to another through temporary visits is a factor [17].

10.8 Project: Effect of Early Detection The following model for Ebola is considered in [10]: I + lJ dS = −βS , dt N dE1 I + lJ = βS − k1 E1 , dt N dE2 = k1 E1 − k2 E2 − fT E2 , dt dI = k2 E2 − (α + γ )I, dt dJ = αI + fT E2 − γr J, dt dR = γ (1 − δ)I + γr (1 − δ)J, dt dD = γ δI + γr δJ, dt

(10.41)

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10 Models for Ebola

where N = S + E1 + E2 + I + J + R. The variables E1 and E2 denote the numbers of latent undetectable and detectable individuals, respectively, I is the number of infectious individuals, J is the number of isolated infective individuals, R and D are numbers of recovered and dead due to the disease. For the parameters, β is the transmission rate, l is the relative transmissibility of isolated individuals, k is the rate of entering the detectable class, k2 is the rate of becoming infective, fT is the rate of being diagnosed, α is the rate of isolation, γ is the recovery rate, and γr is the rate at which individuals are removed from isolation after recovery or disease death.

References 1. Althaus, C.L. (2014) Estimating the reproduction number of Ebola Virus (EBOV) during the 2014 outbreak in West Africa, PLoS Currents: Edition 1. https://doi.org/10.1371/current. outbreaks.91afb5e0f279e7f29e7056095255b288. 2. Barbarossa, M.V., A Denes, G. Kiss, Y. Nakata, G. Rost, & Z. Vizi (2015) Transmission dynamics and final epidemic size of Ebola virus disease dynamics with varying interventions, PLoS One 10(7): e0131398. https://doi.org/10.1371/journalpone.0131398. 3. Bellan, S.E., J.R.C. Pulliam, J. Dushoff, and L.A. Meyers (2014) Ebola control: effect of asymptomatic infection and acquired immunity, The Lancet 384:1499–1500. 4. Bichara, D., Y. Kang, C. Castillo-Chavez, R. Horan and C. Perringa (2015) SI S and SI R epidemic models under virtual dispersal, Bull. Math. Biol. 77: 2004–2034. 5. Blower, S.M. and H. Dowlatabadi (1994) Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. Int Stat Rev. 2: 229–243. 6. Brauer, F. (2019) The final size of a serious epidemic, Bull. Math. Biol. https://doi.org/10.1017/ s11538-018-00549-x 7. Browne, C.J., H. Gulbudak, and G. Webb (2015) Modeling contact tracing in outbreaks with application to Ebola, J. Theor. Biol. 384: 33–49. 8. Butler, D. (2014) Models overestimate Ebola cases, Nature 515: 18. https://doi.org/10.1038/ 515018a. 9. Byrne, J.P. (2008) Encyclopedia of Pestilence, Pandemics, and Plagues, 1 ABC-CLIO, 2008. 10. Chowell, D., C. Castillo-Chavez, S. Krishna, X. Qiu, and K.S. Anderson (2015) Modelling the effect of early detection of Ebola, Lancet Infectious Diseases, February, DOI: http://dx.doi.org/ 10.1016/S1473-3099(14)71084-9. 11. Chowell, G., N.W. Hengartnerb, C. Castillo-Chavez, P.W, Fenimore, and J.M. Hyman (2004) The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda, J. Theor. Biol. 229: 119–126. 12. Chowell, G. and J. M. Hyman (2016) Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases. Springer, 2016. 13. Chowell G., C. Viboud, J.M. Hyman, L. Simonsen (2015) The Western Africa Ebola virus disease epidemic exhibits both global exponential and local polynomial growth rates. PLOS Current Outbreaks, January 21. 14. Chowell, G. and C. Viboud (2016) Is it growing exponentially fast?–impact of assuming exponential growth for characterizing and forecasting epidemics with initial near-exponential growth dynamics Infectious Dis. Modelling 1: 71–78. 15. Chowell, G., C. Viboud, L. Simonsen, & S. Moghadas (2016) Characterizing the reproduction number of epidemics with early sub-exponential growth dynamics, J. Roy. Soc. Interface: https://doi.org/10.1098/rsif.2016.0659.

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16. Colgate, S.A., E. A. Stanley, J. M. Hyman, S. P. Layne, and C. Qualls (1989) Risk behaviorbased model of the cubic growth of acquired immunodeficiency syndrome in the United States. Proc. Nat. Acad. Sci., textbf86: 4793–4797 17. Espinoza, B., V. Moreno, D, Bichara, and C. Castillo-Chavez (2016) Assessing the efficiency of movement restriction as a control strategy of Ebola, In Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases, Springer International Publishing: pp. 123–145. 18. Feng, Z., Z. Zheng, N. Hernandez-Ceron, J.W. Glasser, and A.N. Hill (2016) Mathematical models of Ebola - Consequences of underlying assumptions, Math. Biosc. 277: 89–107. 19. Fisman, D.N., T.S. Hauck, A.R. Tuite, & A.L. Greer (2013) An IDEA for short term outbreak projection: nearcasting using the basic reproduction number, PLOS One 8: 1–8. 20. Fisman, D.N., E. Khoo, & A.R. Tuite (2014) Early epidemic dynamics of the west Africa 2014 Ebola outbreak: Estimates derived with a simple two-parameter model, PLOS currents 6: September 8. 21. Hethcote, H.W. (2000) The mathematics of infectious diseases, SIAM Review 42: 599–653. 22. Hsieh, Y.-H. (2009) Richards model: a simple procedure for real-time prediction of outbreak severity In Modeling and Dynamics of Infectious Diseases, pages 216–236. World Scientific. 23. Khan, A., M. Naveed, M. Dur-e-Ahmad, and M. Imran (2015) Estimating the basic reproductive ratio for the Ebola outbreak in Liberia and Sierra Leone, Infect. Dis. Poverty 4. 24. Khan, A.S, F.K. Tshioko, D.L. Heymann, et al (1999) The reemergence of Ebola hemorrhagic fever, Democratic Republic of the Congo, 1995, J. Inf. Dis. 179: S76–86. 25. King, K., et al (2015) Avoidable errors in the modelling of outbreaks of emerging pathogens, with special reference to Ebola, Proc. R. Soc. B 282: 20150347. http://dx.doi.org/10.1098/rspb. 2015.0347 26. Legrand, J., R.F. Grais, P.Y. Boelle, A.J. Valleron, and A. Flahault (2007) Understanding the dynamics of Ebola epidemics, Epidemiol. Infect 135: 610–621. 27. Lewnard, J.A., M.L.N. Mbah, J.A. Alfaro-Murillo, et al. (2014) Dynamics and control of Ebola virus transmission in Montserrado, Liberia: a mathematical modelling analysis, Lancet Infect. Dis. 14: 1189–1195. 28. Manfredi, P. & A. d’Onofrio (eds.) (2013) Modeling the Interplay between Human Behavior and the Spread of Infectious Diseases, Springer - Verlag, New York-Heidelberg-DordrechtLondon. 29. Meltzer M.I., C.Y. Atkins, S. Santibanez, et al. (2014) Estimating the future number of cases in the Ebola epidemic: Liberia and Sierra Leone, 2014–2015. MMWR Surveill. Summ. 63: 2014–2015. 30. McNeil Jr., D.G. Jr. 9@014) NYT: Using a Tactic Unseen in a Century, Countries Cordon Off Ebola-Racked Areas, August 12, 2014. 31. Nishiura, H and G. Chowell (2014) Early transmission dynamics of Ebola virus disease (EVD), west Africa, March to August 2014, Euro Surveill. 19: 1–6. 32. Nyenswah, T.G., F. Kateh, L. Bawo, M. Massaquoi, M. Gbanyan, M. Fallah, T. K. Nagbe, K. K. Karsor, C. S. Wesseh, S. Sieh, et al. (2016) Ebola and its control in Liberia, 2014–2015, Emerging Infectious Diseases 22: 169. 33. Renshaw, E. (1991) Modelling Biological Populations in Space and Time. Cambridge University Press, Cambridge, 1991. 34. Richards, F. (1959) A flexible growth function for empirical use, J. Experimental Botany 10: 290–301. 35. Rivers, C.M. et al. (2014) Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia, PLoS Current Outbreaks, November 6, 2014. 36. Towers, S., O. Patterson-Lomba, and C. Castillo-Chavez (2014) Temporal variations in the effective reproduction number of the 2014 West Africa Ebola Outbreak, PLoS Currents: Outbreaks 1. 37. Tuite, A.R. & D.N. Fisman (2016) The IDEA model: A single equation approach to the Ebola forecasting challenge, Epidemics, http://dx.doi.org/10.1016/j.epidem.2016.09.001

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38. van den Driessche, P. & J. Watmough (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosc., 180: 29–48. 39. Viboud, C., L. Simonsen, and G. Chowell (2016) A generalized-growth model to characterize the early ascending phase of infectious disease outbreaks, Epidemics 15: 27–37. 40. Webb G., C. Browne, X. Huo, O. Seydi, M. Seydi, and P. Magal (2014) A model of the 2014 Ebola epidemic in west Africa with contact tracing. PLoS Currents, https://doi.org/10.1371/ currents.outbreaks.846b2a31ef37018b7d1126a9c8adf22a. 41. World Health Organization (WHO) (2001) Outbreak of Ebola hemorrhagic fever, Uganda, August 2000 - January 2001. Weekly epidemiological record 2001;76:41–48.

Chapter 11

Models for Malaria

Malaria is one of the most important diseases transmitted by vectors. The vectors for many vector-transmitted diseases are mosquitoes or other insects which tend to be more common in warmer climates. One influence of climate change in coming years may be to extend the regions where mosquitoes can thrive and thus to cause the spread of vector-transmitted diseases geographically.

11.1 A Malaria Model As we have remarked earlier, many of the important underlying ideas of mathematical epidemiology arose in the study of malaria begun by Sir R.A. Ross [11]. Malaria is one example of a disease with vector transmission, the infection being transmitted back and forth between vectors (mosquitoes) and hosts (humans). It kills nearly 1,000,000 people annually, mostly children and mostly in poor countries in Africa. We begin with a basic vector transmission model, namely the model (6.2) of Sect. 6.2 reduced by assuming both hosts and vectors satisfy SI R structure. For simplicity, we assume that there are no disease deaths of either hosts or vectors so that the two populations have constant total sizes Nh , Nv , respectively. The resulting model is Sh′ = Λh − βh Sh

Iv − μh Sh Nv

Iv − (μh + γh )Ih Nv Ih Sv′ = Λv − βv Sv − μv Sv Nh I h − (μv + γv )Iv . Iv′ = βv Sv Nh Ih′ = βh Sh

© Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_11

(11.1)

391

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11 Models for Malaria

There is always a disease-free equilibrium (Nh , 0, Nv , 0), and there may also be an endemic equilibrium with the infective population sizes of both species positive. At an endemic equilibrium, βh Sh Iv = (γh + μh )Ih Nv βv Sv Ih = (γv + μv )Ih   Iv Nh Λh = Sh μh + βh Nv   Ih . Λv = Sv μv + βv Nh In applying the next generation matrix approach to determine the basic reproduction number, we obtain     h 0 βh N 0 γh + μh N v F = . , V = Nv 0 βv N 0 γv + μv h This leads to ⎡

⎢ F V −1 = ⎣

N

0 Nv βv N h γh +μh

⎤

βh Nhv γv +μv ⎥

0

⎦.

This would imply that R0 =

"

βh βv . (γv + μv )(γh + μh )

However, this approach views the transition from host to vector to host as two generations, and it may be more reasonable to say that R0 =

βh βv . (γv + μv )(γh + μh )

(11.2)

With either choice, the transition is at R0 = 1. Since a single infectious host infects Nv βv N h

μh + γh

11.1 A Malaria Model

393

vectors in a completely susceptible vector population, and each of these infects h βh N Nv

μv + γv hosts in a completely susceptible host population. Thus, the number of secondary host infections caused by an infective host (and also the number of secondary vector infections caused by an infective vector) is given by (11.2), and this is a valid choice for R0 . The other choice is also a valid choice and it is essential to check which form is used in a given work. By linearizing the system (11.1) at an equilibrium , it is possible to show that the disease-free equilibrium is asymptotically stable if and only if R0 < 1 and that the endemic equilibrium exists only if R0 > 1 and is asymptotically stable. Because (11.1) is a four-dimensional system, this requires showing that the roots of a fourth degree polynomial have negative real parts, and this is technically complicated. For malaria, with humans as hosts and mosquitoes as vectors, we modify the vector transmission model (6.2) in two ways. Since infected mosquitoes remain infected for life and do not recover, we take γv = 0. Also, βh = bThv ,

βv = bTvh ,

where b is the biting rate, the number of bites made by a mosquito in unit time, Thv is the probability that a bite by an infected mosquito will infect a susceptible human, and Tvh is the probability that a bite by a susceptible mosquito of an infected human will infect the mosquito. This gives the model Sh′ = Λh − bThv Sh

Iv − μh Sh Nv

Iv − (μh + γh )Ih Nv Ih − μv Sv Sv′ = Λv − bTvh Sv Nh I h Iv′ = bTvh Sv − μv Iv . Nh Ih′ = bThv Sh

(11.3)

This is basically the original model of Ross [11], with modifications added by MacDonald [8–10]. It is known as the Ross–MacDonald model, and it has remained the basic malaria model ever since its development. If we write the basic reproduction number in terms of the mosquito biting rate, we obtain R0 =

b2 Thv Tvh . μv (γh + μh )

(11.4)

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11 Models for Malaria

Thus, the basic reproduction number depends on the ratio of vector population size to host population size, increases with the vector population size but decreases with the host population size. For the infection to spread, there must be enough vectors that frequently a single host may be bitten at least twice—once by an infected vector to infect the host and a second time to transmit infection to another vector. Macdonald [10] speaks of a stability index, aTvh /μv , which is very difficult to estimate in any particular situation, but gives some indication of the qualitative behavior of a malaria model. Large values of the stability index indicate “stable malaria,” which suggests that malaria would be endemic, while small values of the stability index in a region suggest that it is more likely that there will be epidemic outbreaks. Estimates of the stability index in various regions have ranged from 0.5 to 5.0. Dr. Ross was awarded the second Nobel Prize in Medicine for his demonstration of the dynamics of the transmission of malaria between mosquitoes and humans. His work received immediate acceptance in the medical community, but his deduction that malaria could be controlled by controlling mosquitoes was dismissed on the grounds that it would be impossible to rid a region of mosquitoes completely and that in any case, mosquitoes would soon re-invade the region. Ross then formulated a mathematical model [11] predicting that malaria outbreaks could be avoided if the mosquito population could be reduced below a critical threshold level, and this is in fact, the first instance of the idea of the basic reproduction number in modeling disease transmission. Field trials supported Ross’s conclusions and led to sometimes brilliant successes in malaria control. A notable example is the draining of swamps in the Galilee region in Israel to reduce mosquito habitat. Unfortunately, the Garki project provides a dramatic counterexample. This project worked to eradicate malaria from a region temporarily. However, people who have recovered from an attack of malaria have a temporary immunity against reinfection. Thus, elimination of malaria from a region leaves the inhabitants of this region without immunity when the campaign ends, and the result can be a serious outbreak of malaria. While the model (11.3) describes the basic properties of malaria quite well, it ignores important aspects that could be added. For example, including an incubation period for the mosquito population would give considerably better quantitative agreement with data. Another extension would include seasonality in mosquito population density. The model (11.3) assumes that infected humans are not subject to further infection, but there is evidence to indicate that “superinfection” may be an important phenomenon. MacDonald’s 1957 model [10] included successive infections waiting to express themselves when the previous infection ends. While this addition does not affect the qualitative behavior, it does affect quantitative results. Immunity to malaria acquired by repeated exposure to malaria has the effect, in regions in which malaria is endemic, of causing most cases of malaria to be in children. Modeling this would require inclusion of age structure, as well as including acquired immunity that is boosted by reinfection. One way to model this would be

11.2 Some Model Refinements

395

to assume that there is an immune period of fixed length following recovery in the absence of exposure but that if a person is exposed again during a fixed period following the loss of immunity the immunity is not lost [7, p. 179]. A description of various extensions of the basic Ross–MacDonald model may be found in [3, 13] Controlling malaria by controlling mosquito populations is a fine theoretical solution. However, mosquitoes adapt very rapidly to insecticides and control methods must be revised frequently in practice. Malaria remains one of the diseases causing the largest number of deaths worldwide.

11.2 Some Model Refinements The model (11.3) is a basic model which has been the standard model for many years. As we have pointed out in the preceding section, there are many possible refinements. In this section, we give a little more detail of some of these refinements.

11.2.1 Mosquito Incubation Periods The basic malaria model (11.3) gives a good description of malaria infection dynamics, but some of its quantitative predictions are quite different from observations. Some of these differences can be resolved by taking into account the incubation period for mosquitoes [10, 11]. The most elementary model including an incubation period assumes an exposed period with an exponential rate of moving from exposed to infectious mosquitoes, Sh′ = Λh − bThv Sh

Iv − μh Sh Nv

Iv − (μh + αh )Ih Nv Ih − μv Sv (t) Sv′ (t) = Λv − bTvh Sv Nh Ih Ev′ (t) = bTvh Sv − (μ + κv )Ev Nh ′ Iv (t) = κEv − μv Iv .

Ih′ = bThv Sh

(11.5)

To determine the basic reproduction number of the model (11.5), we use the next generation matrix method with disease states (Ih , Ev , Iv ). We have ⎡

0

⎢ Nv F = ⎣bTvh N h 0

⎤ h 0 bThv N Nv ⎥ 0 0 ⎦, 0 0

⎡

⎤ μh + γh 0 0 V =⎣ 0 μv + κv 0 ⎦ 0 0 μv

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11 Models for Malaria

This leads to ⎡

⎤ Nh 1 κv h bTvh N Nv μh (μv +κv ) bThv Nv μv ⎥ 0 0 ⎦, 0 0

0

⎢ Nv 1 F V −1 = ⎣bTvh N h μh +γh 0 from which we derive R0 = b2 Tvh Thv

κv . μh (μh + γh )(μv + κv )

In fact, the model (11.3) was modified by assuming that infection in mosquitoes has an incubation period of fixed length τ [10, 11]. This gives a model Sh′ = Λh − bThv Sh

Iv − μh Sh Nv

Iv − (μh + γh )Ih Nv Ih Sv′ (t) = Λv − bTvh Sv (t) − μSv (t) Nh Ih Ih − e−μτ bTvh Sv (t − τ ) Ev′ (t) = bTvh Sv (t) Nh Nh Ih Iv′ (t) = e−μτ bTvh Sv (t − τ ) − μv Iv . Nh Ih′ = bThv Sh

(11.6)

For this model, R0 = e−μτ

b2 Thv Tvh . μv (γh + μh )

As in the preceding section, it would be possible to use the square root as the basic reproduction number.

11.2.2 Boosting of Immunity Another property of malaria is that the immunity obtained by recovery from infection is not permanent. Recovery brings temporary immunity and this immunity is boosted by exposure to further infection. This may be described by the following model. We divide the host population into three compartments, S the number of uninfected members, I the number of infected individuals, and R the number of removed individuals with immunity. We assume that susceptible individuals become infected at a rate h. They then recover at a rate r to become immune. There is a rate of reversion from immune to susceptible γ , chosen so that the average period of

11.2 Some Model Refinements

397

immunity corresponds to an assumption that immunity lasts until the occurrence of a gap of τ years without exposure [1–3]. Then γ (h) =

he−hτ . 1 − e−hτ

(11.7)

To establish the relation (11.7), we let p = e−hτ be the probability of infection after time τ and q = 1 − p be the probability of infection after time τ . Then γ (h), the rate of loss of immunity, is hp/q, and this establishes (11.7). The loss of immunity has been observed frequently. In regions where there have been mosquito control programs that reduced malaria cases, one effect of terminating a control program has been an increase in malaria cases because of the loss of immunity since there was less boosting of immunity. The mean period of immunity may be estimated from observation, and the relation (11.7) gives the rate of reversion to susceptibility as a function of the rate of developing infection.

11.2.3 Alternative Forms for the Force of Infection Several forms for the force-of-infection for malaria models are considered in [4]. We denote the transmission rates from humans to mosquitoes and from mosquitoes to humans by λh = Thv bh (Nh , Nv )

Iv Nv

and

λv = Tvh bv (Nh , Nv )

Ih , Nh

(11.8)

respectively, where bh =

b(Nh , Nv ) , Nh

bv =

b(Nh , Nv ) , Nv

and b(Nh , Nv ) denotes the total number of mosquito bites on humans given by b=

σv Nv σh Nh σv σh = Nv . σv Nv + σh Nh σv (Nv /Nh ) + σh

(11.9)

Depending on the sizes of the human and mosquito populations, the biting rates bh and bv can take several limiting forms, as summarized in Table 11.1.

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Table 11.1 Number of mosquito bites on humans in the force-of-infection functions (11.8) bh General model As Nh → ∞ or Nv → 0 As Nh → 0 or Nv → ∞

bv

σv σh Nv σv (Nv /Nh ) + σh σv Nv Nh σh

σv σh Nh σv (Nv /Nh ) + σh

b σv Nv σh Nh σv (Nv /Nh ) + σh

σv

σv Nv

σh Nh Nv

σh Nh

11.3 *Coupling Malaria Epidemiology and Sickle-Cell Genetics Although the population dynamics of malaria and the population genetics of the sickle-cell genes occur on very different time scales, it is straightforward to develop an appropriate model relating these. The high mortality associated with malaria has led to strong historical selection for resistance, and hence for single major genes conferring resistance in heterozygotes, despite the associated burden borne by homozygotes. The foundation is the classical Ross–MacDonald model for the spread of malaria, expanded to include the relevant genetic structure of the host. Let Sh1 denote the density of uninfected humans of genotype AA. Similarly, Sh2 denotes the population density of genotype AS. Furthermore, let Ih1 and Ih2 represent the population densities of infected individuals of each genotype. We ignore SS individuals; high mortality rates from sickle-cell disease are typical in countries with high transmission rates of falciparum malaria, so these individuals rarely reach reproductive maturity. An extended model including the SS individuals can be studied using similar methods but it is very difficult to interpret the threshold conditions due to the complexity of the model. Finally, let z be the fraction of mosquitoes that are transmitting malaria. The fraction of the AS individuals in the population is w=

Sh2 + Ih2 Nh

where Nh = Sh1 + Ih1 + Sh2 + Ih2 is the total human population density. The frequency of the S gene is denoted by q = w/2 and the frequency of the A gene is denoted by p = 1 − q. Let B(N) denote the human per-capita birth rate, possibly density dependent (e.g., logistic growth), with a constant per-capita natural mortality of μh . To couple ecology and evolution, we make two assumptions. First, we assume that the ratio of mosquitoes to humans is a constant, c. This is a standard assumption in the modeling of malaria (ever since the original Ross–MacDonald model). Any other assumption about variability in the ratio of mosquitoes to humans (Nv /Nh ) would need to be justified.

11.3 *Coupling Malaria Epidemiology and Sickle-Cell Genetics

399

Second, we assume that the fraction of each genotype born into the population Pi is given by P1 = p2 ,

P2 = 2pq.

The transmission of malaria between humans and mosquitoes is governed by some basic epidemiological parameters. The human biting rate is denoted by b, and average life of an infected mosquito is 1/μv . The probability that a human develops a parasitemia from a bite is denoted Thvi ; we assume that Thv1 ≥ Thv2 . The disease induced death rate is denoted by ρi , and we assume that ρ1 ≫ ρ2 . In addition, we consider that AS individuals may die faster than AA individuals from causes other than malaria, and the excess rate of mortality for AS individuals is ν. The probability that a mosquito acquires plasmodium from biting an individual of type i is denoted by Tvhi . The average time until a victim of malaria recovers, denoted by 1/γhi , may be different in AA and AS individuals. The changes in population density of each genotype with each infection status are described by a set of five coupled ordinary differential equations: ′ = P B(N)N − m S − bT czS + γ I , Shi i i hi hvi hi hi hi ′ = bT czS − (m + γ + ρ )I , Ihi hvi hi i hi hi hi

(11.10)

 Ih2  Ih1 − μv z, + bTvh2 z′ = (1 − z) bTvh1 Nh Nh

i = 1, 2,

where m1 = μh and m2 = μh + ν. More detailed analysis of the model (11.10) can be found in [5, 6]. It is both mathematically convenient and biologically relevant to introduce new variables for prevalence of malaria infections in each genotype, xi = ui /N and yi = vi /N, as well as the frequency of the S-gene, w = x2 + y2 = 2q. The equations in the new variables are derived from the original (11.10) using the chain rule. We note for clarification that x1 + y1 + x2 + y2 = 1

and

x1 + y1 = 1 − w.

We also introduce notation to reduce the number of parameters, βhi = bThvi c, βvi = bTvhi , i = 1, 2. Then, we obtain the following system equivalent to (11.10) in the terms that describe important epidemiological, demographic, and population genetic quantities, y1 , y2 , z, w, and Nh :

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11 Models for Malaria

y1′ = βh1 z(1 − w − y1 ) − (m1 + γh1 + ρh1 )y1 − y1 Nh′ /Nh , y2′ = βh2 z(w − y2 ) − (m2 + γh2 + ρh2 )y2 − y2 Nh′ /Nh , z′ = (1 − z)(βv1 y1 + βv2 y2 ) − μv z,

(11.11)

w ′ = P2 B(N) − ρh2 y2 − m2 w − wNh′ /Nh , Nh′ = Nh ((P1 + P2 )B(N ) − m1 (1 − w) − m2 w − ρh1 y1 − ρh2 y2 ) . Although most of the equations assume a general birth function B(N), our detailed mathematical analysis for the specific case in which B(N) is a density dependent per-capita birth function, B(N) = B0 (1 − N/K), where B0 is a constant (the maximum birth rate when population size is small) and K is approximately the density dependent reduction in birth rate. The relevant parameters vary across many orders of magnitude. For example, the demographic parameters (B0 and mi ) and the genetic parameters (ρhi ) are on the order of 1/decades, and the malaria disease parameters (βhi , γhi , βvi , and μv ) are on the order of 1/days. Hence, although the malaria disease dynamics and the changes in genetic composition are two coupled processes, the former occurs on a much faster time scale than the latter. Let mi = ǫ m ˜ i , ρhi = ǫ ρ˜hi , and B0 = ǫ B˜ 0 with ǫ > 0 being small. We can use this fact to simplify the mathematical analysis of the full model with the use of singular perturbation techniques, which allows us to separate the time scales of the different processes. By letting ǫ = 0, we obtain the following system for the fast dynamics: y1′ = βh1 z(1 − y1 − w) − γh1 y1 , y2′ = βh2 z(w − y2 ) − γh2 y2 , z′ = (1 − z)(βv1 y1 + βv2 y2 ) − μv z,

(11.12)

which describes the epidemics of malaria for a given distribution of genotypes determined by w. Here, on the fast time scale, w is considered as a parameter. On the fast time scale, the basic reproduction number of malaria disease can be calculated as the leading eigenvalue of the next generation matrix: R0 = R1 (1 − w) + R2 w,

(11.13)

where Ri = βγhihi βμviv , i = 1, 2 involves parameters associated with malaria √ transmission between mosquitoes and humans of genotype i. In fact, Ri (or Ri ) is the basic reproduction number when the population consists of entirely humans of genotype i. It can be shown that, when R0 < 1, the disease-free equilibrium of the system (11.12) is locally asymptotically stable; and when R0 > 1, the system (11.12) has a unique non-trivial equilibrium E ∗ = (y1∗ , y2∗ , z∗ ) given by

11.3 *Coupling Malaria Epidemiology and Sickle-Cell Genetics

y1∗ =

Qh2 z∗ Qh1 z∗ ∗ (1 − w), y = w, 2 1 + Qh1 z∗ 1 + Qh2 z∗

401

(11.14)

where Qhi = βhi /γhi for i = 1, 2, and z∗ is a solution of the equation k0 z2 + k1 z + k2 = 0,

(11.15)

with k0 = Th1 Th2 + R1 Th2 (1 − w) + R2 Th1 w, k1 = Th1 + Th2 + R1 (1 − Th2 )(1 − w) + R2 (1 − Th1 )w,

(11.16)

k 2 = 1 − R0 . It can be shown that Eq. (11.15) has a positive solution only if R0 > 1. When R0 < 1, note that k2 = 1 − R0 > 0 and k1 > 0; it follows that Eq. (11.15) has no positive solution. Hence, E ∗ is not biologically feasible. If R0 > 1, then k2 = 1 − R0 < 0. It is easy to show that k0 > 0 as 0 < w < 1. Hence, Eq. (11.15) has a unique positive solution z∗ . Let h(z) denote the function of z given by the left-hand side of (11.15). Notice that h(0) = k2 < 0, h(1) = 1 + Th1 Th2 + Th1 + Th2 > 0, and h(z∗ ) = 0. Hence, 0 < z∗ < 1. From (11.14) we also have that 0 < yi∗ < 1, i = 1, 2. It follows that an endemic equilibrium E ∗ = (y1∗ , y2∗ , z∗ ) exists and is unique. For the stability of E ∗ , it can be shown that all eigenvalues of the Jacobian matrix at E ∗ have negative real part if and only if the following expression: λ =:

"

βh1 (1 − w − y1∗ ) βv1 y1∗ + βv2 y2∗ + μv



    βh2 (w − y2∗ ) βv1 (1 − z∗ ) βv2 (1 − z∗ ) + βh1 z∗ + γh1 βv1 y1∗ + βv2 y2∗ + μv βh2 z∗ + γh2

is less than 1 (see [5] for more details). Using the following Equalities: z1∗ =

βv1 y1∗ + βv2 y2∗ γh1 y1∗ , = βv1 y1∗ + βv2 y2∗ + μv βh1 (1 − w − y1∗ )

z2∗ =

γh2 y2∗ βv1 y1∗ + βv2 y2∗ = , ∗ ∗ βv1 y1 + βv2 y2 + μv βh2 (w − y2∗ )

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11 Models for Malaria

and noticing that βhi z∗ + γhi > γhi , i = 1, 2, and that 0 < z∗ < 1, we get λ2
σ1 . This is indeed confirmed by both analytical and numerical studies of the slow system. Figure 11.1a shows a bifurcation diagram of the slow dynamics with σ1 and σ2 being the bifurcation parameters. In Fig. 11.1, B˜ 1∗ is a constant larger than the maximum per-capita birth rate B˜ (=B/ǫ); σ2 = h(σ1 ) is a decreasing function satisfying

Fig. 11.1 Phase portraits (a) of the slow system in the (σ1 , σ2 ) plane. Plots (b)–(e) illustrate the phase portraits of the slow system for (σ1 , σ2 ) in different regions

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11 Models for Malaria

˜ = B˜ and h(B˜ ∗ ) = 0. Some of the results from this bifurcation diagram are h(B) summarized as follows: Case 1: σ2 < σ1 (Positive Fitness) ˜ or B˜ < σ1 < B˜ ∗ and σ2 < h(σ1 ), then there is a unique interior (a) If σ1 ≤ B, equilibrium E∗ = (w∗ , Nh∗ ) that is globally asymptotically stable (g.a.s.), (b) If b˜ < σ1 < B˜ ∗ and σ2 > h(σ1 ), then the population will be wiped out (due to the death rates being too much higher than the “birth” rate) with the fraction of AS individuals tending to a positive constant as t → ∞. Case 2: σ2 > σ1 (Negative Fitness)

The fraction of AS individuals will tend to zero as t → ∞, whereas the total population size will tend to either K (when σ2 is small) or zero (when σ2 is large). In either case, the system (11.18) has neither periodic solutions nor homoclinic loops. An analytic proof of these results can be found in [6]. We point out that Case 1(b) is due to the standard incidence form of infection rate used in the z equation. Similar scenarios have been observed in other population models, and such scenarios may not be present if the mass action form is used. The standard incidence form is more appropriate if the number of contacts is relatively constant, independent of density. Figure 11.1b–e demonstrates some numerical calculations of solutions of the system (11.18) for (σ1 , σ2 ) in different regions. It shows a couple of possible scenarios when σ1 < σ2 . It is interesting to notice that it is possible for the system to have two locally asymptotically stable equilibria. One is the boundary equilibrium at which Nh > 0 and w = 0, and the other is one of the two interior equilibria. The regions of attraction of the two stable equilibria are divided by the separatrix formed by the stable manifold of the unstable interior equilibrium. This type of bi-stability can occur in several different ways. There are also cases when the system (11.18) has none, or one, or two equilibria on the positive w-axis. It shows that, if w(0) is small, i.e., the initial population size of AS individuals is small, then the S-gene will be extinct due to a negative fitness. However, if for some reason (e.g., immigration of AS individuals) w suddenly becomes large (large enough to be on the right side of the separatrix), then the S-gene will be able to establish itself, even though the fitness is negative. These results can be used to study questions related to the evolution of associated traits. We see from above that whether or not the S-gene can invade and establish itself in a population is determined by whether the fitness coefficient is positive or negative. Recall that the fitness is given by the difference σ1 − σ2 , where the total death rate σi is a sum of weighted death rates mi and ρhi (see (11.21)) with the weight Wi given by (11.20). The quantity Wi contains all the malaria transmission parameters.

11.3 *Coupling Malaria Epidemiology and Sickle-Cell Genetics

405

In [12], an extension of model (11.10) is considered by including a class with asymptomatic malaria. It assumes that among all malaria infections, the proportion k is assumed to be symptomatic, while a proportion, 1 − k, is asymptomatic. It is assumed that a proportion f of individuals who experience symptomatic infections receives antimalarial therapy. It is known that sickle-cell traits protect individuals against symptomatic but not against asymptomatic infections [14, 15]. Thus, we let 1 − ζ denote the level of protection against symptomatic infections among the subpopulation of individuals with sickle-cell traits. Furthermore, 1/γh is used to denote the average infective period of symptomatic patients, and the malaria-induced death rate is denoted by ρhi (ρh1 ≫ ρh2 ). Let Mhi denote symptomatically infected or treated individuals of type i, and let Ahi denote asymptomatically infected hosts of type i. For treated individuals, the average infective period of symptomatic patients can be shortened to 1/γhT and disease mortality is reduced to ρhT . With or without treatment, infections do not confer permanent immunity, allowing repeated infections. The model reads dSh1 = P1 B(Nh )Nh − βh Sh1 z + γh (Ih1 + Ah1 ) + γhT Mh1 − μh1 Sh1 , dt dSh2 = P2 B(Nh )Nh − [ζ k + (1 − k)]βh Sh2 z + γh (Ih2 dt + Ah2 ) + γhT Mh2 − μh2 Sh2 , dIh1 dt dI2 dt dMh1 dt dMh2 dt dAh1 dt dAh2 dt dz dt

where Nh =

= k(1 − f )βh Sh1 z − (γh + μh1 + ρh1 )Ih1 , = ζ k(1 − f )βh Sh2 z − (γh + μh2 + ρh2 )Ih2 , = kfβh Sh1 z − (γhT + μh1 + ρhT 1 )Mh1 , = ζ kfβh Sh2 z − (γhT + μh2 + ρhT 2 )Mh2 , = (1 − k)βh Sh1 z − (γh + μh1 )Ah1 , = (1 − k)βh Sh2 z − (γh + μh2 )Ah2 , =

βv (Ih1 + Ih2 + Ah1 + Ah2 + δMh1 + δMh2 )(1 − z) − μv z, Nh (11.22)

2

i=1 Shi

+ Ihi + Mhi + Ahi and B(Nh ) = B0 (1 − Nh /K).

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11 Models for Malaria

Similarly to the analysis of the system (11.10), it can be shown that on the fast time scale, the basic reproduction number of malaria disease can be calculated as the leading eigenvalue of the next generation matrix: Rc =(1 − g)



βv βh μv γh



+g



βv βh (1 − k(1 − ζ )) μv γh



(11.23)

=(1 − g)R1 + gR2 . Here Ri (i = 1, 2) is defined as Ri = Qv Qhi , where Qv =

βv , μv

Qh1 =

βh , γh

and

Qh2 =

βh (1 − k(1 − ζ )) . γh

It can be shown that E0 is locally asymptotically stable if Rc < 1 and unstable if Rc > 1. The model can be used to explore the influence of S-gene frequency on the malaria epidemic. To assess how the frequency of the S-gene g influences the overall endemic level of malaria, we examined how the reproduction number of malaria (Rc ) changes when g is varied (Fig. 11.2). In general, the disease prevalence increases with Rc . It is also worth noting that Rc = (1 − g)R1 + gR2 where R1 and R2 are contributions from non-carriers and carriers of sicklecell disease, respectively (see (11.23)). As a result, R1 is greater than R2 in the presence of the S-gene, and Rc decreases with the frequency of the S-gene (g). This proves that the overall prevalence of malaria infection decreases with an increasing number of individuals who have the sickle-cell trait. However, this effect is stronger on symptomatic infections than on asymptomatic infections (Fig. 11.2). Nevertheless, the relative prevalence of asymptomatic infections among sickle-

Fig. 11.2 Impact of sickle-cell trait on the epidemiology of malaria as the frequency of AS individuals (g) was varied (solid: ζ = 0.6; dashed: ζ = 0.4; dotted: ζ = 0.2). The left figure ∗ + I ∗ )/N ∗ , blue) and asymptomatic ((A∗ + A∗ )/N ∗ , plots the prevalence of symptomatic ((Ih1 h h h2 h1 h2 black) malaria infection for different ζ . The figure on the right plots the relative prevalence of asymptomatic AA individuals and asymptomatic AS individuals compared to all symptomatic ∗ + I ∗ ) in green and A∗ /(I ∗ + I ∗ ) in magenta) individuals (shown as A∗h1 /(Ih1 h2 h2 h1 h2

References

407

Fig. 11.3 The left figure is a plot of the level curves of the control reproduction number (Rc ) when the proportion of symptomatic infection (k) and the proportion of treatment (f ) are varied. The figure on the right is a plot of R0 vs relative efficacy against symptomatic infection among individuals of type 2 compared to individuals of type 1, and the proportion of asymptomatic infection

cell carriers increased with the frequency of the S-gene(Fig. 11.2). Specifically, the relative prevalence of asymptomatic infections with sickle-cell trait, compared to symptomatic infections among both the carriers and non-carriers of sickle-cell disease, increased from 0.12 to 0.59 as the frequency of the S-gene increased from 0.10 to 0.40. This increase occurred partially because the sickle-cell trait does not provide protection against asymptomatic malaria. This pattern was more pronounced when the relative susceptibility of sickle-cell carriers to malaria (ζ ) decreased (Fig. 11.2). Figure 11.2 demonstrates how various factors may influence the disease dynamics. The effects on Rc can be examined using the formula given in (11.23). This is shown in Fig. 11.3, which shows how Rc changes as the probability of treatment (f ) or the proportion of symptomatic infection (k) varies. We observe that the value of Rc was more sensitive to the changes in the proportion of k than to changes in the treatment probability f . Furthermore, when the proportion of symptomatic infection (k) is relatively low, increasing treatment rate is unlikely to be effective in lowering the burden of malaria. Given that the asymptomatic infection of malaria is common in malaria-endemic regions, our result indicates that the effect of treatment might be limited. In addition, strong selection of sickle-cell traits (i.e., lower value of ζ ) is likely to reduce the impact of treatment on reducing the prevalence of malaria.

References 1. Aron, J.L. (1982) Dynamics of acquired immunity boosted by exposure to infection, Math. Biosc. 64: 249–259. 2. Aron, J.L. (1988) Mathematical modeling of immunity to malaria (1988) Math. Biosc. 90: 385–396.

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3. Aron, J.L. & R.M. May (1982) The population dynamics of malaria, in Population Dynamics of Infectious Diseases, R.M. Anderson, ed. Chapman and Hall, London, pp. 139–179. 4. Chitnis, N., J.M.Cushing, & J.M. Hyman (2006) Bifurcation analysis of a mathematical model for malaria transmission, SIAM J. App. Math. 67: 24–45. 5. Feng, Z., D.L. Smith, E.F. McKenzie & S.A. Levin (2004) Coupling ecology and evolution: malaria and the S-gene across time scales, Math. Biosc. 189: 1–19, 6. Feng, Z., Y. Yi, & H. Zhu (2004) Fast and slow dynamics of malaria and the S-gene frequency, Journal of Dynamics and Differential Equations, 16: 869–896. 7. Karlin, S. & H.M. Taylor (1975) A First Course in Stochastic Processes, 2nd ed., Academic Press, New York. 8. MacDonald, G. (1950) The analysis of infection rates in diseases in which superinfection occurs, Tropical Diseases Bull. 47: 907–915. 9. MacDonald, G. (1952) The analysis of equilibrium in malaria, Tropical diseases Bull. 49: 813– 828. 10. MacDonald, G. (1957) The Epidemiology and Control of Malaria, Oxford University Press, London. 11. Ross, R. (1911) The Prevention of Malaria, 2nd ed., (with Addendum), John Murray, London. 12. Shim, E., Z. Feng, C. Castillo-Chavez (2012) Differential impact of sickle cell trait on symptomatic and asymptomatic malaria, Math. Biosc. & Eng., 9: 877–898. 13. Teboh-Ewungkem, M.I., G.A. Ngwa & C.N. Ngonghala (2013) Models and proposals for malaria: A review Math. Pop. Studies 20: 57–81. 14. Vafa M., M. Troye-Blomberg, J. Anchang, A. Garcia & F. Migot-Nabias (2008) Multiplicity of Plasmodium falciparum infection in asymptomatic children in Senegal: relation to transmission, age and erythrocyte variants, Malaria J. 7: 17. 15. Williams T.N. (2006) Human red blood cell polymorphisms and malaria, Curr. Opin. Microbiol. 9: 388–394.

Chapter 12

Dengue Fever and the Zika Virus

12.1 Dengue Fever While there have been cases of probable dengue fever more than 1000 years ago, the first recognized dengue epidemics occurred in Asia, Africa, and North America in the 1780s. There have been frequent outbreaks since then, and the number of reported cases has been increasing rapidly recently. According to the World Health Organization, approximately 50,000,000 people worldwide are infected with dengue. Symptoms may include fever, headaches, joint and muscle pain, and nausea, but many cases are very mild. There is no cure for dengue fever, but most patients recover with rest and fluids. There are at least four different strains of dengue fever, and there is some cross-immunity between strains. Dengue fever is transmitted by the mosquito aedes aegypti, and most control strategies are aimed at mosquito control. Dengue, a re-emerging vector-borne disease, is caused by members of the genus Flavivirus in the family Flaviviridae with four active antigenically distinct serotypes, DENV-1, DENV-2, DENV-3, and DENV-4 [15]. The pathogenicity of dengue can range from asymptomatic, mild dengue fever (DF), to dengue hemorrhagic fever (DHF), and dengue shock syndrome (DSS) [15, 23]. Although infection with a dengue serotype does not usually protect against other serotypes, it is believed that secondary infections with a heterologous serotype increase the probability of DHF and DSS [10, 22]. According to the World Health Organization, 40% of the global population is at risk for dengue infection with an estimate of 50–100 million infections yearly including 500,000 cases of DHF. It has been estimated that about 22,000 deaths, mostly children under 15 years of age, can be attributed to DHF [46]. In the United States, approximately 5% or more of the Key West population in Florida was exposed to dengue during the 2009– 2010 outbreak [11] while the Hawaii Department of Health reported 190 cases

© Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_12

409

410

12 Dengue Fever and the Zika Virus

during the 2015 outbreak on Oahu, the first outbreak since 2011. Since dengue is not endemic in Hawaii, health authorities have suggested that the recent outbreak may have been started by infected visitors [25]. Dengue is highly prevalent and endemic in Southeast Asia, which has experienced a 70% increase in cases since 2004 [29]; Mexico, also an endemic country, reported over a million cases of DF and more than 17,000 cases of DHF [21, 33] during the 2002 outbreak. Dengue is transmitted primarily by the vector Ae. aegypti, which is now found in most countries in the tropics [24, 37]. The secondary vector, Ae. albopictus, has a range reaching farther north than Ae. aegypti with eggs better adapted to subfreezing temperatures [26, 33]. Differences in susceptibility and transmission of dengue infection [3, 27, 42] raise the possibility that some serotypes are either more successful at invading a host population, or more pathogenic, or both [30]. DENV2 is the most associated with dengue outbreaks involving DHF and DSS cases [32, 38, 48], followed by DENV-1 and DENV-3 viruses [4, 24, 32]. While infection with any of the four dengue serotypes could lead to DHF, the rapid displacement of DENV-2 American by DENV-2 Asian genotype has been linked to major outbreaks with DHF cases in Cuba, Jamaica, Venezuela, Colombia, Brazil, Peru, and Mexico [31, 32, 38, 39, 41, 48]. A possible mechanism involved in the dispersal and persistence of DENV-2 in nature is vertical transmission (transovarial transmission) via Ae. aegypti. Advances in molecular biology have been used to show that vertical transmission involving Ae. aegypti and Ae. albopictus is possible in captivity and in the wild [3, 8, 12, 20, 34, 40]. Thus, assessing transmission dynamics and pathogenicity between the DENV-2 American and Asian genotypes’ differences is one of the priorities associated with the study of the epidemiology of dengue. In short, dengue has an increasing recurrent presence putting a larger percentage of the global population at risk of dengue infection, a situation that has become the norm due to the growth of travel and tourism between endemic and non-endemic regions. The potential role of vertical transmission in dengue endemic regions or in fluctuating environments has been explored in [1, 18, 36]. The role of host movement has also been explored in the context of dengue [2] in a formulation that does not account for the effective population size. The model (6.2) in Chapter 6 is a generic model for vector-transmitted diseases. For any specific disease, it is necessary to modify this model to incorporate properties of the disease not included in the generic model. Suppose we assume that the mosquito population is in equilibrium, so that the mosquito birth rate is βv Nv . We assume also vertical transmission for mosquitoes. The birth rate of mosquitoes is μv Nv , of which μv Iv are born to infective mother mosquitoes, and we assume that a fraction q of these are born infective. Then a model describing the dynamics of DENV-2 is given by the following system of differential equations:

12.1 Dengue Fever

411

Sh′ = μh Nh − βh Sh

Iv − μh Sh Nv

Iv − (ηh + μh )Eh Nv Ih′ = ηh Eh − (γ + μh )Ih Ih Sv′ = μv (Nv − qIv ) − βv Sv − μv Sv Nh Ih Ev′ = βv Sv − (ηv + μv )Ev Nh Iv′ = qμv Iv + ηv Ev − μv Iv . Eh′ = βh Sh

(12.1)

This model is the same as the basic vector transmission model (6.2) in Chapter 6 except that the birth rate of susceptible hosts is now μh Nh and vertical transmission of hosts is added. In the absence of selection, that is, differences in birth and death rate and in the absence of vertical transmission, the model (12.1) turns out to be equivalent to a model considered by Chowell et al. in [13]. Model (12.1) is well defined supporting a sharp threshold property, namely, the disease dies out if the basic reproduction number R0 is less than unity and persists whenever R0 > 1.

12.1.1 Calculation of the Basic Reproduction Number We calculate the basic reproduction number for the model (12.1) in two stages, as we did for the model (6.2) in Sect. 6.2. In the first stage, an infective mosquito infects humans at a rate βNh /Nv for a time 1/μv , producing βNh /Nv μv infected humans per mosquito. In the second stage, an infective human infects mosquitoes, at a rate βv Nv /Nh for a time 1/(μh + γ ). This produces βv Nv /Nh (γ + μh ) infected mosquitoes, of whom a fraction ηv /(ηv + μv ) proceeds to become infective. The net result of these two stages is Rv =

ηv ηv 1 ηh 1 ηh 1 1 βv Nv = βh βv Nh μh + γ ηv + μv ηh + μh μv μh + γ ηv + μv ηh + μh μv (12.2)

infected vectors. In addition, an infective mosquito produces Rd = qμv

(12.3)

412

12 Dengue Fever and the Zika Virus

infective mosquitoes through vertical transmission, giving a total basic reproduction number R 0 = R v + Rd = βh βv

(12.4)

ηh 1 ηv 1 + qμv . μh + γ ηv + μv ηh + μh μv

We could also calculate the basic reproduction number by using the next generation matrix approach [45]. If we interpret only human infections as new infections and consider vector infections as transitions, we would obtain the same result.

12.2 A Model with Asymptomatic Infectives Many cases of dengue are very mild and may not be reported. We can incorporate this in a model by assuming that a fraction p of exposed members become infective while the remainder of the exposed class go to an asymptomatic stage with lower infectivity and possibly more rapid recovery. We have already described a model with such transitions in influenza models in Sect. 9.2. We consider a model including this structure, namely Sh′ = μh Nh − βh Sh Eh′ = βh Sh

Iv − μh Sh Nv

Iv − (ηh + μh )Eh Nv

Ih′ = pηh Eh − (γ + μh )Ih

A′h

(12.5)

= (1 − p)ηh Eh − (κ + μh )Ah

Sv′ = μv (Nv − qIv ) − βv Sv Ev′ = βv Sv

Ih + δAh − μv Sv Nh

Ih + δAh − (ηv + μv )Ev Nh

Iv′ = qμv + ηv Ev − μv Iv . Here, δ is the infectivity reduction factor for asymptomatics, and κ is the recovery rate for asymptomatics.

12.2.1 Calculation of the Basic Reproduction Number We calculate the basic reproduction number for the model (12.5) in two stages, as we did for the model (12.1) in the previous section.

12.2 A Model with Asymptomatic Infectives

413

In the first stage, an infective mosquito infects humans at a rate βNh /Nv for a time 1/μv , producing βNh /Nv μv infected humans per mosquito. A fraction ηh /(ηh + μh ) of these proceed to an infective stage, with pβh

ηh Nh μv Nv ηh + μh

going to Ih and (1 − p)βh

ηh Nh μv Nv ηh + μh

going to Ah . In the second stage, an infective human infects mosquitoes, at a rate βv Nv /Nh for a time 1/(μh + γ ). An asymptomatic infects mosquitoes at a rate δβv Nv /Nh for a time 1/(μh + κ). A fraction ηv /(ηv + μv ) of each of these groups develop into infective mosquitoes. Thus, the second stage produces ηv Nv βv Nh ηv + μv



p δ(1 − p) + μh + γ μh + κ



infected mosquitoes. The net result of these two stages is Nh ηh ηv Nv Rv = βh βv μv Nv ηh + μh Nh ηv + μv



p δ(1 − p) + μh + γ μh + κ



(12.6)

infected mosquitoes. In addition, an infective mosquito produces Rd = qμv

(12.7)

infective mosquitoes through vertical transmission, giving a total basic reproduction number R0 = Rv + Rd .

(12.8)

Again, we could also calculate the basic reproduction number by using the next generation matrix approach [45]. If we interpret only human infections as new infections and consider vector infections as transitions, we would obtain the same result, but if we interpreted both human and vector infections we would obtain a different version of the basic reproduction number R∗ =

   1 Rd + Rd2 + 4Rv . 2

414

12 Dengue Fever and the Zika Virus

12.3 The Zika Virus The Zika virus, a mosquito borne arbovirus, was first identified in Uganda in 1947. Similar to the dengue and chikungunya viruses, Zika is primarily spread by the mosquito, Aedes aegypti. Recent outbreaks of Zika disease have occurred in Yap Island in the Pacific in 2007 [16] and in French Polynesia in 2013–2014 [28]. Since 2015, Zika has spread through much of South America, Central America, and the Caribbean, where the Aedes aegypti species is endemic, recently reaching pandemic levels in 2016. While the reasons for the explosive spread of the disease in the Americas are still unclear, the rapid urbanization in countries with under-developed infrastructure for surveillance and vector control probably plays a role. Zika disease is usually asymptomatic, and typically is mild even with a clinical presentation. Its symptoms are similar to those of dengue and chikungunya virus infections. However, the disease has been linked to an apparent increased risk of the neurological disorder Guillain–Barré syndrome, and also to neonate microcephaly. The latter is of particular concern, because pregnant women may not even know they have been infected, and the damage to their unborn infants may result in subsequent lifelong disabilities. There is currently no vaccine or specific treatment for Zika infection, leaving control of the vector populations and avoidance through the use of mosquito repellents as the only means to control the spread of the disease. For the Zika virus, it has been established that in addition to vector transmission of infection there may also be direct transmission through sexual contact. The Zika virus is the first example of an infection that can be transferred both directly and through a vector, and it is important to include direct transmission (in this case sexual transmission) in a model. Estimation of the basic reproduction number is particularly difficult for Zika because of the difficulty in estimating the relative importance of transmission through vectors and direct transmission through sexual contact. A vaccine is being developed for the Zika virus, and analysis of the question of whether this vaccine can control an outbreak is contained in [44].

12.4 A Model with Vector and Direct Transmission We approach the question of formulating a model for the Zika virus by beginning with the basic vector transmission model (6.2) and adding direct host to host transmission. We add to the model (6.2) a term αSh NIhh describing a rate α of movement from S to E. Also, we consider a single outbreak model, and omit demographic terms and vertical transmission in the host population. This leads to the following model [9]: Sh′ = −βh Sh Eh′ = βh Sh

Ih Iv − αSh Nv Nh

Iv Ih + αSh − ηh Eh Nv Nh

12.4 A Model with Vector and Direct Transmission

415

Ih′ = ηh Eh − γ Ih

(12.9)

Sv′ = μv Nv − μv Sv − βv Sv Ev′ = βv Sv

Ih Nh

Ih − (μv + ηv )Ev Nh

Iv′ = ηv Ev − μv Iv . The rate α is an average over the human population; if transmission is possible only from male to female this is incorporated into α. To calculate the basic reproduction number R0 , we use the same direct approach as that used in Section 6.2. If there is sexual transmission, this operates independent of the host–vector interaction, and produces α cases in unit time for a time 1/γ , adding a simple term α/γ to the reproduction number R0 = βh βv

α ηv + . μv γ (μv + ηv ) γ

(12.10)

ηv , μv γ (μv + ηv )

(12.11)

We define Rv = βh βv

the vector transmission reproduction number, and Rd =

α , γ

(12.12)

the direct transmission reproduction number, so that R 0 = Rv + Rd . If we use the next generation matrix approach, using the same approach as that used in Sect. 12.1.1, we form the matrix product KL = F V −1 with ⎤ ⎡ ⎡ ⎤ h 0 α 0 βh N ηh 0 0 0 Nv ⎥ ⎢ ⎢−ηh γ 0 0⎥ ⎢0 0 0 0 ⎥ ⎥. F =⎢ ⎥, V = ⎢ Nv ⎣ 0 0 ⎦ 0 0 μv + ηv 0 ⎦ ⎣0 βv Nh

0

0

0

0

0

0

−ηv

μv

Then the next generation matrix with large domain is ⎡

α γ

α γ

⎢ 0 0 ⎢ KL = ⎢ N v 1 N ⎣βv N γ βv Nv γ1 h h 0

0

ηv Nh 1 h βh N Nv μv (μv +ηv ) βh Nv μv

0

0

0

0

0

0

⎤

⎥ ⎥ ⎥. ⎦

416

12 Dengue Fever and the Zika Virus

The next generation matrix K is the 2 × 2 matrix K=



α γ Nv 1 βv N h γ

 ηv h βh N Nv μv (μv +ηv ) . 0

The positive eigenvalue of this matrix is " 1 α2 α + + 4Rv λ= 2γ 2 γ2    1 2 = Rd + Rd + 4Rv . 2 We may calculate that λ = 1 if and only if Rv + Rd = 1. We now have two potential expressions for the basic reproduction number, namely Rv + Rd , with Rv and Rd given by (12.11) and (12.12) respectively, and R∗ =

   1 Rd + Rd2 + 4Rv . 2

Different expressions are possible for the next generation matrix and these may lead to different expressions for the basic reproduction number. This is shown in [14]. The expression Rv + Rd appears to us to be a more natural form than R ∗ , and we choose to use this for the basic reproduction number. It can be obtained from the following expression for the next generation matrix. We consider only human infections as new infections, and take ⎤ ⎡ ⎡ ⎤ h 0 0 0 ηh 0 α 0 βh N Nv ⎢ ⎢−η ⎥ γ 0 0⎥ ⎥ ⎢0 0 0 0 ⎥ ⎢ h F =⎢ ⎥. ⎥, V = ⎢ Nv ⎣0 0 0 0 ⎦ ⎣ 0 −βv Nh μv + ηv 0 ⎦ 000 0 0 0 −ηv μv Then

V

−1

⎡

1 ηh

0

⎢ 1 ⎢ [2pt] γ1 γ =⎢ N N 1 ⎢ βv v βv Nhv γ (μηv v+ηv ) ⎣ Nh γ (μv +ηv ) ηv ηv Nv Nv βv N βv N h μv γ (μv +ηv ) h μv γ (μv +ηv )

0 0

⎤ 0 ⎥ 0⎥ ⎥. 0⎥ ⎦

1 μv +ηv ηv 1 μv (μv +ηv ) μ

12.4 A Model with Vector and Direct Transmission

417

Since only the first row of F has non-zero entries, the same is true of F V −1 , and from this we can deduce that the only non-zero eigenvalue of F V −1 is the entry in the first row, first column of F V −1 , and this is ηv α + βh βv = Rd + Rv = R0 . γ μv γ (μv + ηv ) If we use this somewhat unorthodox approach to the next generation matrix for the model (6.2), we obtain the form Rv , with no square root, for the reproduction number. We now have two viable expressions for the basic reproduction number, namely R ∗ and R0 , both derived from a next generation matrix approach but with different separations. We have chosen to use R0 for the basic reproduction number because it is more readily interpreted as a number of secondary infections. Other sources, including [19], use R ∗ . In studying data for epidemic models that include vector transmission it is absolutely vital to specify exactly which form is being used for the basic reproduction number.

12.4.1 The Initial Exponential Growth Rate In order to determine the initial exponential growth rate from the model, a quantity that can be compared with experimental data, we linearize the model (12.9) about the disease-free equilibrium S = Nh , Eh = Ih = 0, Sv = Nv , Ev = Iv = 0. If we let y = Nh − S, z = Nv − Sv , we obtain the linearization Iv + αIh Nv I v Eh′ = βh Nh + αIh − ηh Eh Nv Ih′ = ηh Eh − γ Ih Ih z′ = −μv z + βv Nv Nh Ih Ev = βv Nv − (μv + ηv )Ev Nh Iv′ = ηv Ev − μv Iv . y ′ = βh Nh

(12.13)

The corresponding characteristic equation is ⎡

⎤ h −λ 0 α 0 0 βh N Nv ⎢ ⎥ h ⎢ 0 −(λ + ηh ) ⎥ α 0 0 βh N Nv ⎢ ⎥ ⎢ 0 ⎥ ηh −(λ + γ ) 0 0 0 ⎥ = 0. det ⎢ Nv ⎢ 0 ⎥ 0 0 0 βv Nh −(λ + μv ) ⎢ ⎥ ⎢ ⎥ Nv 0 βv Nh 0 −(λ + μv + ηv ) 0 ⎣ 0 ⎦ 0 0 0 0 ηv −(λ + μv )

418

12 Dengue Fever and the Zika Virus

We can reduce this equation to a product of two factors and a fourth degree polynomial equation ⎡

⎤ h −(λ + ηh ) α 0 βh N Nv ⎢ ⎥ ηh −(λ + γ ) 0 0 ⎢ ⎥ λ(λ + μv )det ⎢ ⎥ = 0. Nv 0 βv N −(λ + μ + η ) 0 ⎣ ⎦ v v h 0 0 ηv −(λ + μv ) The initial exponential growth rate is the largest root of this fourth degree equation, which reduces to g(λ) =(λ + ηh )(λ + γ )(λ + μv + ηv )(λ + μv ) − βh βv ηv ηv

(12.14)

− ηh α(λ + μv )(λ + μv + ηv ) = 0. The largest root of this equation is the initial exponential growth rate, and this may be measured experimentally. If the measured value is ρ, then from (12.14) we obtain (ρ + ηh )(ρ + γ )(ρ + μv + ηv )(ρ + μv ) − βh βv ηh ηv −ηh α(ρ + μ + v)(ρ + μv + ηv ) = 0.

(12.15)

From (12.15) we can see that ρ = 0 corresponds to R0 = 1, confirming that our calculated value of R0 has the proper threshold behavior. Equation (12.15) determines the value of βh βv , and we may then calculate R0 , provided we know the value of α. However, this presents a major problem. In [19] it is suggested that the contribution of sexual disease transmission is small, based on estimates of sexual activity and the probability of disease transmission. Since the probability of sexual transmission of a disease depends strongly on the particular disease, this estimate is quite uncertain. Estimates based on a possible imbalance between male and female disease prevalence are also quite dubious. Most Zika cases are asymptomatic or quite light but the risks of serious birth defects means that diagnosis of Zika is much more important to women than to men. If there are more female than male cases, it is not possible to distinguish between additional cases caused by sexual contact and cases identified by higher diagnosis rates. To the best of our knowledge, there is not yet a satisfactory resolution of this problem. What would be required would be another quantity which can be determined experimentally and can be expressed in terms of the model parameters. In the absence of further information, all we can accomplish is to estimate reproduction numbers for various choices of α and βh βv that satisfy (12.15). We use the parameter values [43] obtained for the 2015 Zika outbreak in Barranquilla, Colombia, including an analysis of the exponential rise in confirmed Zika cases identified by the Colombian SIVIGILA surveillance system up to the end of December, 2015. κ = 1/7

γ = 1/5

ηv = 1/9.5

μv = 1/13,

12.5 A Second Zika Virus Model Table 12.1 Reproduction number values

419 α 0 0.1 0.2 0.3 0.4 0.413

βh βv 0.243 0.184 0.125 0.0665 0.0076 0

Rd 0 0.5 1.0 1.5 2.0 2.065

Rv 4.86 3.69 2.51 1.335 0.152 0

R0 4.86 4.19 3.51 2.835 2.152 2.065

R∗ 2.185 2.187 2.16 2.13 2.074 2.065

S∞ 14 24 45 79 166 185

and the estimated measurement ρ = 0.073. With these values we have 11βh βv + 6.48α = 2.676. To satisfy this equation, we must have 0 ≤ α ≤ 0.413. We then calculate R0 and R ∗ for several values of α in this range, assuming population sizes of 1000 humans and 4000 mosquitoes. We obtain the results summarized in Table 12.1. We observe that R ∗ is not very sensitive to changes in the direct contact rate while R0 is quite sensitive to changes in α. We have also shown the results of simulations of the model (12.9) showing how the epidemic size depends on α. These simulations suggest that the epidemic final size does vary considerably, and without some way of estimating how many disease cases arise from direct contact we are unable to estimate the epidemic final size.

12.5 A Second Zika Virus Model A model for the Zika virus with somewhat more detail than the model (12.9) has been described in [19]. This model includes the assumption that many Zika cases are asymptomatic, and has two infectious stages acute and convalescent. The model takes the form κEh + Ih1 + τ Ih2 Iv Sh′ = −abSh − βSh N Nh h   Iv κEh + Ih1 + τ Ih2 ′ − νh Eh Eh = θ abSh + βSh Nh Nh ′ Ih1 = νh Eh − γh1 Ih1 ′ Ih2 = γh1 − γh2   Ih2 Iv κEh + Ih1 + τ Ih2 ′ − γh Ah Ah = (1 − θ ) bSh + βSh Nh Nh ′ Rh = γh2 + γh Ah Sv ηAh + Ih1 Sv′ = μv Nv − μv Sv − ac Nh ηAh + Ih1 ′ − (μv + νv )Ev Ev = acSv Nh Iv′ = νv Ev − μv Iv .

(12.16)

420

12 Dengue Fever and the Zika Virus

Table 12.2 Model parameters Parameter a b c η β κ τ θ m 1/νh 1/νv 1/γh1 1/γh2 1/γh 1/μv

Description Mosquito biting rate Transmission probability, vector to human Transmission probability, human to vector Transmission probability, asymptomatic humans to vector Transmission probability, human to human Relative transmission probability, exposed to infective Relative transmission probability, convalescent to asymptomatic Proportion of symptomatic infections Ratio of mosquitoes to humans Human incubation period (days) Mosquito incubation period (days) Acute phase duration (days) Convalescent period duration (days) Asymptomatic infection duration (days) Mosquito lifetime (days)

Value 0.5 0.4 0.5 0.1 0.1 0.6 0.3 18 5 5 10 5 20 7 14

The reproduction numbers for this model are βκθ βθ βτ θ + + , Rd = νh γh1 γh2

Rv =



a 2 bcθ a 2 bηcθ + νh μv γh1 μv



ηv . ηv + μv

Parameter values were chosen to fit vector transmission data for Brazil, El Salvador, and Colombia up to February 2016. For direct (sexual) transmission of infection, parameters were chosen with an assumed probability of transmission of infection per sexual contact, but this assumption might be questionable (Table 12.2). With these parameter values, this model yielded estimates of Rd = 0.136,

Rv = 3.842,

R0 = 3.973,

R ∗ = 2.055.

so that

However, there are indications that the number of cases of Zika due to sexual contacts may be considerably higher. Perhaps, the value of β should be larger.

12.6 Project: A Dengue Model with Two Patches

421

12.6 Project: A Dengue Model with Two Patches When dengue fever invades a location, it tends to move from one patch to another. To describe this, we consider a model for DENV-2 in two separate patches, with movement between the patches. The single patch model (12.1) is the building block for the two-patch model. Within each patch, in the absence of host mobility, dengue dynamics are modeled via the system (12.1). We consider an epidemic model in two patches, one of which has a significantly larger contact rate, with short term travel between the two patches. The total population resident in each patch is constant. We follow a Lagrangian perspective, that is, we keep track of each individual’s place of residence at all times [6, 17]. It is assumed that vectors do not move between patches since the vectors Ae. aegypti and Ae. albopictus do not travel more than few tens of meters over their lifetime [2, 47]; moving 400–600 meters at most [7, 35], respectively. In short, we neglect vector dispersal. Thus we consider two patches, with total resident population sizes N1 and N2 respectively, each population being divided into susceptibles, infectives, and removed members. Si and Ii denote the number of susceptibles and infectives respectively who are residents in Patch i, regardless of the patch in which they are present. The host residents of Patch 1, population size Nh,1 , spends, on average, p11 proportion of its time in their own Patch 1 and p12 proportion of its time visiting Patch 2. Residents of Patch 2, population of size Nh,2 , spend p22 proportion of their time in Patch 2 while spending p21 = 1 − p22 visiting Patch 1. Thus, at time t, the effective population in Patch 1 is p11 Nh,1 + p21 Nh,2 and the effective population in Patch 2 is p12 Nh,1 + p22 Nh,2 . The susceptible population of Patch 1 (Sh,1 ) could be infected by a vector in either Patch 1 (Iv,1 ) or Patch 2 by (Iv,2 ), depending on which patch they are located in at the time of infection. Thus, the dynamics of the susceptible population in Patch 1 are given by S˙h,1 = μh Nh,1 − βh Sh,1

2 j =1

aj p1j

Iv,j − μh Sh,1 . p1j Nh,1 + p2j Nh,2

(12.17)

The effective infectious population in Patch 1 is p11 Ih,1 +p21 Ih,2 and, consequently, the proportion of infectious individuals in Patch 1, is p11 Ih,1 + p21 Ih,2 . p11 Nh,1 + p21 Nh,2 The dynamics of susceptible mosquitoes in Patch 1 are modeled as follows: p11 Ih,1 + p21 Ih,2 − μv Sv,1 . S˙v,i = μv (Nv,i − qIv,i ) − βv Sv,1 p11 Nh,1 + p21 Nh,2

(12.18)

422

12 Dengue Fever and the Zika Virus

Question 1 Show that the dynamics of DENV-2, with the host moving between patches, is given by the system ′ Sh,i = μh Nh,i − βh Sh,i ′ Eh,i = βh Sh,i

2 j =1

aj pij

2

aj pij

j =1

Iv,j − μh Sh,i p1j Nh,1 + p2j Nh,2

Iv,j − (μh + ηh )Eh,i p1j Nh,1 + p2j Nh,2

′ Ih,i = ηh Eh,i − (μh + γi )Ih,i

2

(12.19)

j =1 pj i Ih,j

′ − μv Sv,i Sv,i = qμv (Nv,i − qIv,i ) − βv Sv,i 2 pki Nh,k k=1

2 j =1 pj i Ih,j ′ − (μv + ηv )Ev,i Ev,i = βv Sv,i 2 k=1 pki Nh,k ′ Iv,i = ηv Eh,i + (1 − q)μv Iv,i , i = 1, 2.

Question 2 Determine the basic reproduction number of the model (12.19). Question 3 Obtain a pair of final size relations for the model (12.19). The results of this project, together with appropriate data, can be used to estimate the spread of dengue from one community to another [5].

12.7 Exercises 1.∗ Use the next generation matrix to calculate the basic reproduction number of the model (12.1). 2.∗ Use the next generation matrix to calculate the basic reproduction number of the model (12.5).    3. If Rv and Rd are two non-negative numbers, show that 12 Rd + Rd2 + 4Rv is equal to 1 if and only if Rd + Rv = 1.

References 1. Adams, B. and M. Boots (2010) How important is vertical transmission in mosquitoes for the persistence of dengue? Insights from a mathematical model, Epidemics 2: 1–10 2. Adams, B. and D.D. Kapan (2009) Man bites mosquito: understanding the contribution of human movement to vector-borne disease dynamics, PLos One 4(8); e6373. https://doi.org/10. 1371/journal.pone.0006763.

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3. Arunachalam, N., S.C. Tewari, V. Thenmozhi,R. Rajendran, R. Paramasivan, R. Manavalan, K. Ayanar and B.K. Tyagi (2008) Natural vertical transmissionof dengue viruses by Aedes aegyptiin Chennai, Tamil Nadu, India, Indian J Med Res 127: 395–397. 4. Balmaseda, A., S.N. Hammond, L. Perez, Y. Tellez, S.I. Saborio, J.C. Mercado, J.C., R. Cuadra, J. Rocha, M.A. Perez, S. Silva et al (2006) Serotype-specific differences in clinical manifestations of dengue, Am. J. of tropical medicine and hygiene 74: 449. 5. Bichara, D., S.A. Holechek, J. Velasco-Castro„ A.L. Murillo and C. Castillo-Chavez (2016) On the dynamics of dengue virus type 2 with residence times and vertical transmission, to appear 6. Bichara, D., Y. Kang, C. Castillo-Chavez, R. Horan and C. Perringa (2015) SI S and SI R epidemic models under virtual dispersal, Bull. Math. Biol. 77 (2015):2004–2034. 7. Bonnet, D.D. and D.J. Worcester (1946) The dispersal of Aedes albopictus in the territory of Hawaii, Am J.of tropical medicine and hygiene, 4: 465–476. 8. Bosio, C.F., R.E.X.E. Thomas, P.R. Grimstad and K.S. Rai (1992) Variation in the efficiency of vertical transmission of dengue-1 virus by strains of Aedes albopictus (Diptera: Culicidae, J. Medical Entomology 29: 985–989. 9. Brauer, F., S.M. Towers, A. Mubayi, & C, Castillo-Chavez (2016) Some models for epidemics of vector - transmitted diseases, Infectious Disease Modelling 1: 79–87. 10. Burke, D.S., A. Nisalak, D.E. Johnson and R.M.N. Scott (1988) A prospective study of dengue infections in Bangkok, Am. J. of Tropical Medicine and Hygiene 38: 172., 11. CDC (2010) CDC, Report Suggests Nearly 5 Percent Exposed to Dengue Virus in Key West, Url = http://www.cdc.gov/media/pressrel/2010/r100713.htm, 12. Cecílio, A.B. E.S. Campanelli, K.P.R. Souza, L.B. Figueiredo and M.C. Resende (2009) Natural vertical transmission by Stegomyia albopicta as dengue vector in Brazil, Brazilian Journal of Biology 69: 123–127. 13. Chowell, G., P. Diaz-Duenas, J.C. Miller, A. Alcazar-Velasco, J.M. Hyman, P.W. Fenimore, and C. Castillo-Chavez (2007) Estimation of the reproduction number of dengue fever from spatial epidemic data, Math. Biosc. 208: 571–589. 14. Cushing, J.M. and O. Diekmann (2016) The many guises of R0 (a didactic note), J. Theor. Biol. 404: 295–302. 15. Deubel, V., R.M. Kinney and D.W. Trent (1988) Nucleotide sequence and deduced amino acid sequence of the nonstructural proteins of dengue type 2 virus, Jamaica genotype: comparative analysis of the full-length genome. Virology 65: 234–244. 16. Duffy, M.R., T.H. Chen, W.T. Hancock, A.M. Powers, J.L. Kool, R.S. Lanciotti, et al. (2009) Zika virus outbreak on Yap Island, federated states of Micronesia, NEJM 360:2536–2543. 17. Espinoza, B., Moreno, V., Bichara, D., Castillo-Chavez, C. (2016) Assessing the Efficiency of Movement Restriction as a Control Strategy of Ebola. In Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases, 2016, 123–145, Springer International Publishing. 18. Esteva, L. and C. Vargas (2000) Influence of vertical and mechanical transmission on the dynamics of dengue disease, Math. Biosc. 167: 51–64. 19. Gao, D., Y. Lou, D. He, T.C. Porco, Y. Kuang, G. Chowell, S. Ruan (2016) Prevention and control of Zika as a mosquito - borne and sexually transmitted disease: A mathematical modeling analysis, Sci. Rep. 6, 28070, https://doi.org/10.1038/srep28070. 20. Gunther, J., J.P. Martínez-Muñoz, D.G. Pérez-Ishiwara and J. Salas-Benito (2007) Evidence of vertical transmission of dengue virus in two endemic localities in the state of Oaxaca, Mexico, Intervirology 50: 347–352. 21. Guzman, G, G. Kouri, Gustavo (2003) Dengue and dengue hemorrhagic fever in the Americas: lessons and challenges. J. Clinical Virology 27: 1–13. 22. Halstead, S.B, S. Nimmannitya and S.N. Cohen (1970) Observations related to pathogenesis of dengue hemorrhagic fever. IV. Relation of Yale Journal of Biology and Medicine 42: 311. 23. Halstead, S. B, N.T.Lan, T.T. Myint, T.N. Shwe, A. Nisalak S. Kalyanarooj, S. Nimmannitya, S. Soegijanto, D.W. Vaughn and T.P. Endy (2002) Dengue hemorrhagic fever in infants: research opportunities ignored, Emerging infectious diseases 8: 1474–1479.

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24. Harris, E., E. Videa, L. Perez, E. Sandoval, Y. Tellez, M. Perez, R. Cuadra, J. Rocha, W. Idiaquez, and R.E. Alonso et al (2000) Clinical, epidemiologic, and virologic features of dengue in the 1998 epidemic in Nicaragua, Am. J. of tropical medicine and hygiene 63: 5. 25. State of Hawaii (2015) Dengue outbreak 2015, Department of Health http://www.health. hawaii.gov/docd/dengue-outbreak-2015/. 26. Hawley, W.A, P. Reiter, R.S. Copeland, C.B. Pumpuni and G.B. Craig Jr (1987) Aedes albopictus in North America: probable introduction in used tires from northern Asia, Science 236: 1114. 27. Knox, T.B, B.H. Kay, R.A. Hall and P.A. Ryan (2003) Enhanced vector competence of Aedes aegypti (Diptera: Culicidae) from the Torres Strait compared with mainland Australia for dengue 2 and 4 viruses, J. Medical Entomology 40: 950–956. 28. Kucharski, A.J., S. Funk, R.M. Egge, H-P. Mallet, W.J. Edmunds and E.J. Nilles (2016) Transmission dynamics of Zika virus in island populations: a modelling analysisof the 2013–14 French Polynesia outbreak, PLOS Neglected tropical Diseases DOI 101371 29. Kwok, Y. (2010) Dengue Fever Cases Reach Record Highs, in Time, Sept. 24, 2010. 30. Kyle, J.L. and E. Harris (2008) Global spread and persistence of dengue, Ann. Rev. of Microbiology. 31. Lewis, J.A., G.J. Chang, R.S. Lanciotti, R.M. Kinney, L.W. Mayer and D.W. Trent (1993) Phylogenetic relationships of dengue-2 viruses, Virology 197: 216–224. 32. Montoya, Y.,S. Holechek, O. Cáceres, A. Palacios, J. Burans, C. Guevara, F. Quintana, V. Herrera, E. Pozo, E. Anaya, et al (2003) Circulation of dengue viruses in North-Western Peru, 2000–2001, Dengue Bulletin 27 33. Morens, D.M. and A.S. Fauci (2008) Dengue and hemorrhagic fever: A potential threat to public health in the United States JAMA 299: 214. 34. Murillo, D., S.A. Holechek, A.L. Murillo, F. Sanchez, and C. Castillo-Chavez (2014) Vertical Transmission in a Two-Strain Model of Dengue Fever, Letters in Biomathematics 1: 249–271. 35. Niebylski, M.L. and G.B. Craig Jr (1994) Dispersal and survival of Aedes albopictus at a scrap tire yard in Missouri, Journal of the American Mosquito Control Association, 10: 339–343. 36. Nishiura, H. (2006) Mathematical and statistical analyses of the spread of dengue, Dengue Bulletin 30: 51. 37. Reiter, P. and D.J. Gubler (1997) Surveillance and control of urban dengue vectors, Dengue and dengue hemorragic fever : 45–60. 38. Rico-Hesse, R., L.M. Harrison, R.A. Salas, D. Tovar, A. Nisalak, C. Ramos, J. Boshell, M.T.R. de Mesa, H.M.R. Nogueira and A.T. Rosa (1997) Origins of dengue-type viruses associated with increased pathogenicity in the Americas, Virology 230: 344–251. 39. Rico-Hesse, R., L.M. Harrison, A. Nisalak, D.W. Vaughn, S. Kalayanarooj, S. Green, A.L. Rothman and F.A. Ennis (1998) Molecular evolution of dengue type 2 virus in Thailand, Am.J. of Tropical Medicine and Hygiene 58: 96. 40. Rosen, L., D.A. Shroyer, R.B. Tesh, J.E. Freier and J.C. Lien (1983) Transovarial transmission of dengue viruses by mosquitoes: Aedes albopictus and Aedes aegypti Am. J. tropical medicine and hygiene 32: 1108–1119. 41. Sittisombut, N., A. Sistayanarain, M.J. Cardosa, M. Salminen, S. Damrongdachakul, S. Kalayanarooj, S. Rojanasuphot, J. Supawadee and N. Maneekarn (1997) Possible occurence of a genetic bottleneck in dengue serotype 2 viruses between the 1980 and 1987 epidemic seasons in Bangkok, Thailand, Am. J. of Tropical Medicine and Hygiene 57: 100. 42. Tewari, S.C., V. Thenmozhi, C.R. Katholi, R. Manavalan, A.Munirathinam and A. Gajanana (2004) Dengue vector prevalence and virus infection in a rural area in south India, Tropical Medicine and International Health 9: 499–507. 43. Towers, S., F. Brauer, C. Castillo-Chavez, A.K.I. Falconer, A. Mubayi, C.M.E. Romero-Vivas (2016) Estimation of the reproduction number of the 2015 Zika virus outbreak in Barranquilla, Columbia, Epidemics 17: 50–55. 44. Valega-Mackenzie, W. and K. Rios-Soto (2018) Can vaccination save a Zika virus epidemic?, Bull.Math. Biol. 80:598–625.

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Part III

More Advanced Concepts

Chapter 13

Disease Transmission Models with Age Structure

13.1 Introduction Age is one of the most important characteristics in the modeling of populations and infectious diseases. Because age groups frequently mix heterogeneously it may be appropriate to include age structure in epidemiological models. While there are other aspects of heterogeneity in disease transmission models, such as behavioral and spatial heterogeneity, age structure is one of the most important aspects of heterogeneity in disease modeling. As references for the mathematical details, we suggest [1, Chapters 9 and 13], [21, Part III], [23, Chapter 19], [28, Section 10.9], [32, Chapter 22]. Some applications may be found in [2, 4, 9, 13, 15, 21, 24, 25, 27, 34, 35]. Childhood diseases, such as measles, chicken pox, and rubella, are spread mainly by contacts between children of similar ages. More than half of the deaths attributed to malaria are in children under 5 years of age due to their weaker immune systems. This suggests that in models for disease transmission in an age structured population it may be advisable to allow the contact rates between two members of the population to depend on the ages of both members. Sexually transmitted diseases (STDs) are spread through partner interactions with pair-formations, and the pair formations are age-dependent in most cases. For example, most AIDS cases occur in the group of young adults. More instances of age dependence may be found in [1]. A description of age structured disease transmission requires an introduction to age structured population models. In fact, the first models for age structured populations [26] were designed for the study of disease transmission in such populations. Therefore, we begin with a brief description of some aspects of age structured population models.

© Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_13

429

430

13 Disease Transmission Models with Age Structure

In this chapter, we discuss some structured models that have been used in ecology and epidemiology. Some topics are included in projects. References to the literature indicate where more detailed discussions can be found. We have chosen applications that do not demand a sophisticated mathematical background and that yet give a feeling for some active areas of research.

13.2 Linear Age Structured Models It was a physician, Lieut. Col. A.G. McKendrick (1926), who first introduced age into the description of the dynamics of a one-sex population. The linear theory is based on the McKendrick equation [26], which is most often referred to as the Von Foerster (1959) equation since McKendrick’s earlier work was not well known. Fortunately, the contributions of McKendrick have been made available widely to the scientific community [21]. The McKendrick model assumes that the female population can be described by a density function ρ(a, t) of chronological age a a and time t, both to be measured on the same scale. Then, N (a, t) = 0 ρ(σ, t)dσ is∞the number of individuals of ages in the interval [0, a] at time t, and P (t) = 0 ρ(σ, t)dσ is the total population size at time t. Let β(a) ≥ 0 denote the per-capita age specific fertility rate. Then the total number of births at time t is  ∞ B(t) = β(a)ρ(a, t)da, (13.1) 0

and the total number of births in the interval [t, t + h](h > 0) is given by 

t

t+h  ∞ 0

β(a)ρ(a, t)dads =



t+h

B(s)ds. t

Let μ(a) ≥ 0 denote the per-capita age specific death rate. Then the probability that an individual will survive from birth to age a is π(a) = e−

a

μ(α)dα

0

.

Note that, for each s ∈ [t, t + h], the total number of individuals of age less than or a+s equal to a+s who die at time t +s is 0 μ(σ )ρ(σ, t +s)dσ . Thus, the total number of individuals of age less than or equal to a + u − t who die at time u ∈ [t, t + h] is given by (letting u = s + t): 

t

t+h  a+u−t 0

μ(σ )ρ(σ, u)dσ du =



0

h  a+s 0

μ(σ )ρ(σ, t + s)dσ ds.

13.2 Linear Age Structured Models

431

Assume also that the only changes in the population size from time t to t + h are due to births and deaths. Then N(a + h, t + h) − N (a, t) =



t+h

B(s)ds −

t



0

h  a+s 0

μ(σ )ρ(σ, t + s)dσ ds. (13.2)

The next step is to divide (13.2) by h and let h → 0. In this process, we will need to make use of differentiation of an integral. The basic rule for this is that if F (a, x) =



a

f (ξ, x)dξ, 0

then the partial derivatives of F are given by Fa (a, x) = f (a, x),

Fx (a, x) =



a

fx (ξ, x)dξ. 0

Since N(a+h, t +h)−N (a, t) = [N(a+h, t +h)−N (a, t +h)]+[N (a, t +h)−N (a, t)], the limit as h → 0 of the left side of (13.2) divided by h is Na (a, t) + Nt (a, t) = ρ(a, t) +



a

ρt (α, t)dα.

0

Thus, we obtain the following from the left-hand side: N(a + h, t + h) − N (a, t + h) + N (a, t + h) − N (a, t) h→0  ha lim

= Nt (a, t) + Na (a, t) =

0

(13.3)

ρt (σ, t)dσ + ρ(a, t).

The limit as h → 0 of the right-hand side of (13.2) divided by h, again using differentiation of the integral, leads to B(t) −



a

μ(σ )ρ(σ, t)dσ.

(13.4)

0

Combining (13.3)–(13.4), we obtain 

0

a

ρt (σ, t)dσ + ρ(a, t) = B(t) −



a

μ(σ )ρ(σ, t)dσ. 0

(13.5)

432

13 Disease Transmission Models with Age Structure

Differentiation of (13.5) with respect to a yields ρt (a, t) + ρa (a, t) = −μ(a)ρ(a, t).

(13.6)

Assume that the initial age distribution as ρ(a, 0) = ρ0 (a) and that the gender ratio is one to one. Note that ρ(0, t) also represents the total number of births at time t, i.e., ρ(0, t) = B(t). Then, we arrive at the following partial differential equation with initial and boundary conditions: ρt (a, t) + ρa (a, t) = −μ(a)ρ(a, t),

ρ(0, t) =

ρ(a, 0) =

∞

β(a)ρ(a, t)da = B(t),

(13.7)

0 ρ0 (a).

13.3 The Method of Characteristics System (13.7) can be solved by the method of characteristics. That is, we can solve for ρ(a, t) along the characteristics lines a = t + c, where c is a constant. Let ωc denote the cohort function tracking through time of those individuals whose age is c at t = 0, which is defined as: ωc (t) = ρ(t + c, t),

for t > tc

where tc = max(0, −c). Note that ∂ρ da ∂ρ dt ∂ρ ∂ρ dωc = + = + , dt ∂a dt ∂t dt ∂a ∂t thus, (13.7) can be rewritten as dωc (t) = −μ(t + c)ωc (t), dt

t ≥ tc ,

whose solution is   t  μ(s + c)ds , ωc (t) = ωc (tc ) exp − tc

for t > tc .

(13.8)

13.4 Equivalent Formulation as an Integral Equation Model

433

Therefore, the solution to (13.7) can be written as:   t  ⎧ ⎪ ⎪ ρ (a − t) exp − μ(s + a − t)ds , a≥t ⎨ 0    0a ρ(a, t) = ⎪ ⎪ ⎩B(t − a) exp − μ(s)ds , a < t.

(13.9)

0

Note that B(t) is given in (13.1), which depends on ρ(a, t).

13.4 Equivalent Formulation as an Integral Equation Model Substituting (13.9) into (13.1) yields B(t) =



t 0

β(a)B(t − a)e−

a 0

μ(s)ds

da +



t

∞

ρ0 (a − t)e−

t 0

μ(a−t+s)ds

da. (13.10)

Let F (t) =



t

∞

ρ0 (a − t)e−

t 0

μ(a−t+s)ds

da,

K(a) = β(a)e−

a 0

μ(s)ds

= β(a)π(a).

Then, substituting the expressions for F and K into (13.10) and making a change of variable a = t − u, we obtain the renewal equation: B(t) = F (t) +



t

0

K(t − u)B(u)du,

(13.11)

which was initially derived by Sharpe and Lotka [31]. Using the notation of convolution, we can rewrite Eq. (13.11) in the form B(t) = F (t) + (K ∗ B)(t).

(13.12)

∞ Let fˆ(p) = 0 e−ps f (s)ds denote the Laplace transform of the function f (t). Then, from Eq. (13.12) we have ˆ ˆ ˆ B(p) = Fˆ (p) + B(p) K(p). ˆ Solving for B(p) we get ˆ B(p) =

Fˆ (p) . ˆ 1 − K(p)

434

13 Disease Transmission Models with Age Structure

ˆ ˆ Because K(p) and Fˆ (p) are analytic functions, the properties of B(p) are deterˆ mined by the roots of 1 − K(p), or solutions of ˆ K(p) =



∞ 0

β(a)π(a)e−pa da = 1

(13.13)

for p ∈ C. Equation (13.13) is called the Lotka characteristic equation for the renewal equation (13.11). Kˆ ˆ < 0 we know that K(p) decreases monotonically. Note For p ∈ R, from ddp ˆ that, since K(p) → 0 as p → ∞, Eq. (13.13) has a unique solution, which we denote by p∗ , with the property that ˆ K(0) >1 ˆ K(0) < 1.

p∗ > 0 if p∗ < 0 if

ˆ For p = u + iv ∈ C, note that K(p) = 

0

∞

β(a)π(a)e−ua cos(va)da = 1,

∞ 0



0

β(a)π(a)e−(u+iv)a da = 1. Thus,

∞

β(a)π(a)e−ua sin(va)da = 0.

From |cos(va)| ≤ 1 and Kˆ ′ (u) < 0 for all u ∈ R, we know that u ≤ p∗ . Therefore, ˆ ˆ p∗ is the dominant root (greatest modulus) of K(p). That is, all solutions of K(p) = ∗ 1 have real part less than or equal to p . It can be shown that ∗

B(t) = B0 ep t (1 + Ω(t)),

(13.14)

where B0 > 0 and limt→∞ Ω(t) = 0 (see, e.g., [22]). Therefore, the sign of p∗ determines the asymptotic behavior of the solution B(t), and thus of ρ(a, t). From (13.9) and (13.14), we know that ρ(a, t) has the following asymptotic property: lim ρ(a, t) = lim B0 ep

t→∞

t→∞

∗ (t−a)

π(a).

(13.15)

Define ˆ R = K(0) =



∞

β(a)π(a)da,

0

which is the expected number of offspring for each individual during a lifetime. The results above show that B(t) → 0 (∞) as t → ∞ if and only if R < 1 (> 1). ∞ Recall that P (t) = 0 ρ(a, t)da denotes the total number of individuals at time t. A stable age distribution (or persistent age distribution) is defined to be a solution ρ(a, t) for which ρ(a, t)/P (t) is independent of time t. It follows from (13.15) that

13.5 Equilibria and the Characteristic Equation

435

asymptotically every solution given in (13.9) tends to a stable age distribution as t → ∞.

13.5 Equilibria and the Characteristic Equation For system (13.7), a steady-state, or an equilibrium distribution, ρ ∗ (a), is a timeindependent solution. That is, ρ ∗ (a) satisfies the equations: dρ ∗ (a) = −μ(a)ρ ∗ (a), da  ∞ ρ ∗ (0) = β(a)ρ ∗ (a)da.

(13.16)

0

For the linear system (13.16), the only equilibrium is ρ ∗ (a) = 0, and the linearization at this equilibrium is the same as (13.7). For the stability of ρ ∗ (a) = 0, consider the separable solution of the form ρ(a, t) = A(a)ept

(13.17)

with A(a) > 0. The stability of ρ ∗ (a) is determined by the sign of p. From the second equation in (13.7) we have A(0) =



∞ 0

β(a)A(a)da =: B1 .

(13.18)

Substitution of (13.17) into the first equation in (13.7) yields dA(a) = −(p + μ(a))A(a). da Solving for A(a) and and using A(0) = B1 we have: A(a) = B1 π(a)e−pa ,

a

(13.19)

where π(a) = e− 0 μ(s)ds . Using the definition of B1 in (13.18) with A(a) replaced by the expression in (13.19), we obtain: B1 =



0

∞

β(a)A(a)da =



∞ 0

B1 β(a)π(a)e−pa da.

436

13 Disease Transmission Models with Age Structure

Since B1 = 0, dividing both sides of the above equation by B1 we obtain the following equation which p must satisfy:  ∞ β(a)π(a)e−pa da = 1. (13.20) 0

System (13.20) is the same characteristic equation as (13.13). ˆ defined Note that the left-hand side of (13.20) is the same as the function K(p) ˆ in (13.13). Thus, p > 0 (=, < 0) if and only if R = K(0) > 1 (=, < 1). Therefore, the steady-state ρ ∗ (a) = 0 is locally asymptotically stable if and only if R < 1.

13.6 The Demographic Model with Discrete Age Groups Partition the age interval into a finite number n of subintervals [a0 , a1 ), [a1 , a2 ), . . ., [an−1 , an ), where a0 = 0 and an ≤ ∞. Denote the  ai number of individuals with ages in interval [ai−1 , ai ] by Hi (t), so that Hi (t) = ai−1 ρ(t, a)da, i = 1, . . . , n. Then integrating the partial differential equation in (13.7) from a0 to a1 , we have dH1 (t) + ρ(t, a1 ) − ρ(t, a0 ) + dt



a1 a0

μ(a, P )ρ(t, a)da = 0.

(13.21)

Assume that individuals with ages in each interval have the same vital rates such that β(a, P ) = βi , μ(a, P ) = μi , for a in [ai−1 , ai ], i = 1, . . . , n. Here βi and μi are age-independent, but may be density-dependent. Then, in the age interval [0, a1 ], we have ρ(t, 0) =

n

βi Hi (t),

1



a1 a0

μ ρ(t, a)da = μ1 H1 (t),

which leads to n

dH1 = βi Hi − (m1 + μ1 )H1 . dt

(13.22)

1

Here m1 is the progression rate from groups 1 to 2, defined by m1 = ρ(t, a1 )/H1 (t), and we assume it is time-independent. Integrating the partial differential equation in (13.24) from ai−1 to ai for 2 ≤ i ≤ n, we have dHi = mi−1 Hi−1 − (mi + μi ) Hi , dt

i = 2, . . . , n,

(13.23)

where mi is the age progression rate from groups i to i +1, and we let mn = 0. Then the system in (13.7) is reduced to a system of n ordinary differential equations.

13.7 Nonlinear Age Structured Models

437

13.7 Nonlinear Age Structured Models In the models considered in previous sections, the birth and death rates were linear, and this implies that total population size either grows exponentially (if q > 0), dies out exponentially (if q < 0), or remains constant (if q = 0). In studying models without age structure, we considered situations in which populations have a carrying capacity that is approached as t → ∞. In order to allow this possibility for age-structured models we must assume some non-linearity. We now consider the possibility of birth and death moduli of the form β(a, P (t)) and μ(a, P (t)), depending on total population size. A variant, which can be developed by analogous methods, would be to allow the birth modulus (and possibly also the death modulus) to depend on the density ρ(a, t). We will consider only the continuous case because the methods of linear algebra methods used to treat the linear discrete case have no direct adaptation to the nonlinear discrete model. If the birth and death moduli are allowed to depend on total population size, the description of the model must include P (t). Thus, the full model is now ρa (a, t) + ρt (a, t)+ μ(a, P (t))ρ(a, t) = 0 ∞

β(a, P (t))ρ(a, t)da B(t) = ρ(0, t) = 0  ∞ P (t) = ρ(a, t)da

(13.24)

0

ρ(a, 0) = φ(a).

We can transform the problem just as in the linear case by integrating along characteristics. Using the notation μ∗ (α) = μ(α, P (t − a + α)), the same calculations lead to  a ∗ B(t − a)e− 0 μ (α)dα a ρ(a, t) = ∗ φ(a − t)e− a−t μ (α)dα

for t ≥ a

for t < a.

(13.25)

Substituting these expressions into (13.24), we obtain a pair of functional equations for B(t) and P (t), whose solution gives an explicit solution for ρ(a, t), namely B(t) = b(t) + P (t) = p(t) +



t

β(a, P (t))e−

0



0

t

e−

a 0

μ∗ (α)dα

a 0

μ∗ (α)dα

B(t − a)da,

B(t − a)da,

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13 Disease Transmission Models with Age Structure

where b(t) = p(t) =



∞

t



t

∞

β(a, P (t))φ(a − t)e− φ(a − t)e−

a

a

μ∗ (α)dα

a−t

μ∗ (α)dα

a−t

da,

da.

∞ It is reasonable to assume that 0 φ(a)da < ∞ and that the functions β(a, P ) and μ(a, P ) are continuous and non-negative. Under these hypotheses it is easy to verify that b(t) and p(t) are continuous and non-negative and tend to zero as t → ∞ with b(0) > 0 and p(0) > 0. It is possible to prove that a unique continuous non-negative solution exists on 0 ≤ t < ∞ if supa≥0,P ≥0 β(a, P ) < ∞. This model is due to Gurtin and MacCamy [18]. If ρ(a, t) = ρ(a) is an equilibrium age distribution, then P (t) =



∞ 0

ρ(a)da =: P

and

B(t) =



∞ 0

β(a, P )ρ(a)da =: B

are constant. In this case, the McKendrick equation becomes an ordinary differential equation ρ ′ (a) + μ(a, P )ρ(a) = 0,

ρ(0) = B,

whose solution is ρ(a) = Be−

a 0

μ(α,P )dα

.

Let π(a, P ) = e−

a 0

μ(α,P )dα

(13.26)

denote the probability of survival from birth to age a when the population size is the constant P . Then

From B =

∞ 0

ρ(a) = Bπ(a, P ).

(13.27)

β(a, P )ρ(a)da and (13.27), we obtain  ∞ β(a, P )π(a, P )da. B=B 0

Thus, for an equilibrium age distribution with birth rate B, the population size P must satisfy the equation R(P ) =



∞ 0

β(a, P )π(a, P )da = 1,

(13.28)

13.8 Age-Structured Epidemic Models

439

in which case the birth rate is given by B = ∞ 0

P . π(a, P )da

The quantity R(P ) defined in (13.28) is called the reproduction number, which represents the expected number of offspring that an individual has over a lifetime when the total population size is P . Example 1 Suppose that β is independent of age and is a function of P only. Then  ∞  ∞ B(t) = β(P (t))ρ(a, t)da = β(P (t)) ρ(a, t)da (13.29) 0 0 = P (t)β(P (t)). If we define g(P ) = Pβ(P ), then the problem is reduced to a single functional equation for P (t):  t  a ∗ P (t) = p(t) + e− 0 μ (α)dα g(P (t − a))da 0

together with this explicit formula for B(t) given in (13.29).

13.8 Age-Structured Epidemic Models Consider an age-structured SI R model. Let s(t, a), i(t, a), r(t, a) represent the age densities of susceptible, infective, and removed members, respectively, so that the total numbers of individuals in the respective classes are  ∞  ∞  ∞ S(t) = s(a, t)da, I (t) = i(a, t)da, R(t) = r(a, t)da. 0

0

0

The age density of the total population is ρ(a, t) = s(a, t) + i(a, t) + r(a, t), and the total population size is P (t) = S(t) + I (t) + R(t) =



∞

ρ(a, t)da.

0

Then, the model is described by the following system of equations: st (a, t) + sa (a, t) = − (μ(a) + λ(a, t)) s(a, t) it (a, t) + ia (a, t) = λ(a, t)s(a, t) − (μ(a) + γ (a)) i(a, t), rt (a, t) + ra (a, t) = −μ(a)r(a, t) + γ (a)i(a, t),

(13.30)

440

13 Disease Transmission Models with Age Structure

where λ(a, t) is the per-capita infection rate, which may have various forms depending on the assumption about mixing. One possible form focuses on intracohort mixing: λ(a, t) = c(a)i(a, t), corresponding to the assumption that infection can be transmitted only between individuals of the same age. Another possible form is the intercohort mixing : λ(a, t) =



∞

b(a, α)i(α, t)dα,

0

where b(a, α) denotes the rate of infection from contacts between an infective of age α and a susceptible of age a. For intercohort mixing, it is common to assume that the mixing function has a separable form: b(a, α) = b1 (a)b2 (α). Other parameter functions in (13.30) are μ(a) and γ (a), which represent the agedependent per-capita natural death and recovery rates. The initial conditions are s(a, 0) = s0 (a),

i(a, 0) = i0 (a),

r(a, 0) = 0,

(13.31)

where s0 (a) and i0 (a) are the initial distributions of susceptibles and infectives, respectively. The boundary conditions involve the birth or renewal condition. Under the assumption that all newborns are susceptible, the total number of newborns is s(0, t) =



∞

β(a, P (t))ρ(a, t)da,

(13.32)

0

with i(0, t) = r(0, t) = 0.

13.8.1 *Age-Dependent Vaccination in Epidemic Models In many cases, age-structured models are more appropriate to consider than models without an age structure. For example, if transmission is highly age-dependent due to a higher activity in certain age groups, it might be useful to identify the age groups to target for more effective disease control strategies such as vaccination. The model and results presented below are taken from [10]. It concerns age-dependent vaccination strategies for TB. Divide the population into susceptible, vaccinated, latent, infective, and treated classes, where s(t, a), v(t, a), l(t, a), i(t, a), and j (t, a) denote the associated density functions with these respective epidemiological age-structure classes, and

13.8 Age-Structured Epidemic Models

441

n(t, a) = s(t, a) + v(t, a) + l(t, a) + i(t, a) + j (t, a). Assume that all newborns are susceptible and that the mixing between individuals is proportional to their age-dependent activity level. Assume also that an individual may become infected only through contact with infectious individuals, that vaccination is partially effective (i.e., vaccinated individuals can become infected again but with a reduced susceptibility), that only susceptibles will be vaccinated. The dynamics are governed by the following initial boundary value problem: 

 ∂ ∂ s(t, a) = −β(a)c(a)B(t)s(t, a) − μ(a)s(t, a) − ψ(a)s(t, a), + ∂t ∂a   ∂ ∂ v(t, a) = ψ(a)s(t, a) − μ(a)v(t, a) − δβ(a)c(a)B(t)v(t, a), + ∂t ∂a   ∂ ∂ l(t, a) = β(a)c(a)B(t)(s(t, a) + σj (t, a) + δv(t, a)) + ∂t ∂a −(k + μ(a))l(t, a),   ∂ ∂ i(t, a) = kl(t, a) − (r + μ(a))i(t, a), + ∂t ∂a   ∂ ∂ j (t, a) = ri(t, a) − σβ(a)c(a)B(t)j (t, a) − μ(a)j (t, a), + ∂t ∂a (13.33) where  ∞ i(t, a ′ ) B(t) = p(t, a, a ′ )da ′ , n(t, a ′ ) 0 with initial and boundary conditions: s(t, 0) = Λ, v(t, 0) = l(t, 0) = i(t, 0) = j (t, 0) = 0, s(0, a) = s0 (a), v(0, a) = v0 (a), l(0, a) = l0 (a), i(0, a) = i0 (a), j (0, a) = j0 (a). The function p represents the mixing between susceptible individuals of age a and infectious individual of age a ′ , and is assumed to have the form of proportionate mixing: c(a ′ )n(t, a ′ ) . p(t, a, a ′ ) =  ∞ 0 c(u)n(t, u)du

Λ is the recruitment/birth rate (assumed constant); β(a) is the age-specific (average) probability of becoming infected through contact with infectious individuals; c(a) is the age-specific per-capita contact/activity rate; μ(a) is the age-specific per-capita natural death rate (all of these functions are assumed to be continuous and to be

442

13 Disease Transmission Models with Age Structure

zero beyond some maximum age); k is the per-capita rate at which individuals leave the latent class by becoming infectious; r is the per-capita treatment rate; σ and δ are the reductions in risk due to prior exposure to TB and vaccination, respectively, 0 ≤ σ ≤ 1, 0 ≤ δ ≤ 1; and p(t, a, a ′ ) gives the probability that an individual of age a has contact with an individual of age a ′ given that it has a contact with a member of the population. Proportionate mixing has also been used in other studies including [3, 8, 11, 16, 20]. Using the approach of [11], we assume that p(t, a, a ′ ) = p(t, a ′ ) as explicitly described above. The initial age distributions are assumed to be known and to be zero beyond some maximum age. The model (13.33) is well-posed and the proof is similar to that found in [8]. Notice that n(t, a) satisfies the following equations: 

 ∂ ∂ n(t, a) = −μ(a)n(t, a), + ∂t ∂a n(t, 0) = Λ, n(0, a) = n0 (a) = s0 (a) + v0 (a) + l0 (a) + i0 (a) + j0 (a). Using the method of characteristic curves we can solve for n explicitly: n(t, a) = n0 (a)

F (a) H (a − t) + ΛF (a)H (t − a), F (a − t)

where a

F (a) = e− 0 μ(s)ds , H (s) = 1, s ≥ 0; H (s) = 0, s < 0. Hence, n(t, a) → ΛF (a), (a) p(t, a) →  ∞c(a)F =: p∞ (a), t → ∞. c(b)F (b)db

(13.34)

0

Introducing the fractions u(t, a) =

v(t, a) l(t, a) i(t, a) s(t, a) , w(t, a) = , x(t, a) = , y(t, a) = , n(t, a) n(t, a) n(t, a) n(t, a)

z(t, a) =

j (t, a) , n(t, a)

we get a simplified system of (13.33):

13.8 Age-Structured Epidemic Models

443



 ∂ ∂ + u(t, a) = −β(a)c(a)B(t)u(t, a) − ψ(a)u(t, a), ∂a   ∂t ∂ ∂ w(t, a) = ψ(a)u(t, a) − δβ(a)c(a)B(t)w(t, a), + ∂t ∂a   ∂ ∂ x(t, a) = β(a)c(a)B(t)(u(t, a) + δw(t, a) + σ z(t, a)) − kx(t, a), + ∂a   ∂t ∂ ∂ + y(t, a) = kx(t, a) − ry(t, a), ∂t ∂a   ∂ ∂ z(t, a) = ry(t, a) − σβ(a)c(a)B(t)z(t, a), + ∂t ∂a ∞ B(t) = 0 y(t, a)p(t, a)da,

c(a)n(t, a) p(t, a) =  ∞ , 0 c(u)n(t, u)du u(t, 0) = 1, w(t, 0) = x(t, 0) = y(t, 0) = z(t, 0) = 0.

(13.35)

Let Fψ (a) denote the probability that a susceptible individual has not been vaccinated at age a. Then Fψ (a) = e−

a 0

ψ(b)db

.

The system (13.35) has the infection-free steady state u(a) = Fψ (a), w(a) = 1 − Fψ (a), x(a) = y(a) = z(a) = 0, n(a) = ΛF (a).

(13.36)

To study the local stability of the infection-free equilibrium, we linearize (13.35) about (13.36) and consider exponential solutions of the form x(t, a) = X(a)eλt , y(t, a) = Y (a)eλt , B(t) = B0 eλt + O(e2λt ), where B0 =



∞ 0

Y (a)p∞ (a)da

(13.37)

is a constant and p∞ (a) is as in (13.34). Then the linear parts of the x and y equations in (13.35) are of the form d X(a) = β(a)c(a)B0 Vψ (a) − kX(a), da d λY (a) + Y (a) = kX(a) − rY (a), da

λX(a) +

444

13 Disease Transmission Models with Age Structure

where Vψ (a) = Fψ (a) + δ(1 − Fψ (a)).

(13.38)

An expression for Y (a) can be obtained by solving the above system: Y (a) = B0



a



k β(α)c(α) e(λ+k)(α−a) − e(λ+r)(α−a) Vψ (α)dα. r −k

0

(13.39)

From (13.37) and (13.39) we get B0 = B 0



0

∞ a 0

 k p∞ (a)β(α)c(α) e(λ+k)(α−a) − e(λ+r)(α−a) Vψ (α)dαda. r −k (13.40)

By changing the order of integration, introducing τ = a −α, and dividing both sides by B0 (because B0 = 0) in (13.40) we get the characteristic equation 1=



0

∞ ∞ 0

:= G(λ).

 k p∞ (α + τ )β(α)c(α) e−(λ+k)τ − e−(λ+r)τ Vψ (α)dτ dα r −k (13.41)

Now we are ready to define the net reproduction number as R(ψ) = G(0); i.e.,  ∞ ∞

 k R(ψ) = p∞ (α + τ )β(α)c(α) e−kτ − e−rτ Vψ (α)dτ dα. r − k 0 0 (13.42) Noticing that G′ (λ) < 0, lim G(λ) = 0, λ→∞

lim G(λ) = ∞,

λ→−∞

we know that (13.41) has a unique negative real solution λ∗ if, and only if, G(0) < 1, or R(ψ) < 1. Also, (13.41) has a unique positive (zero) real solution if R(ψ) > 1 (R(ψ) = 1). To show that λ∗ is the dominant real part of roots of G(λ), we let λ = x + iy be an arbitrary complex solution to (13.41). Note that 1 = G(λ) = |G(x + iy)| ≤ G(x), indicating that ℜλ ≤ λ∗ . It follows that the infection-free steady state is locally asymptotically stable if R(ψ) < 1, and unstable if R(ψ) > 1. This establishes the following result.

13.8 Age-Structured Epidemic Models

445

Theorem 13.1 Let R(φ) be the reproduction number defined in (13.42). (a) The infection-free steady state (13.36) is locally asymptotically stable if R(ψ) < 1 and unstable if R(ψ) > 1. (b) There exists an endemic steady state of (13.35) when R(ψ) > 1.

13.8.2 Pair-Formation in Age-Structured Epidemic Models Sexually transmitted diseases (STDs) spread through sexual activities between partners. The pair-formation, or mixing, is one of the key terms in modeling of STDs [19, 20]. Assume that the function describing pair formation in the model is separable, which makes the mathematical analysis more tractable. However, it has been shown that the assumption of a separable pair formation function is equivalent to assuming a total proportionate or random partnership formation [5–8, 12]. We briefly explain it as follows. Let ρ(t, a, a ′ ) be the pair formation or mixing, which is the proportion of partners with age a ′ that an individual of age a has at time t. Let r(t, a) be the average number of partners that an individual of age a has per unit of time, and let T (t, a) be the total number of individuals of age a at time t. Then the function ρ(t, a, a ′ ) has the following properties: 1) 0 ≤ ρ(t, a, a ′ ) ≤ 1,  ∞ 2) ρ(t, a, a ′ )da ′ = 1, 0

3) ρ(t, a, a ′ )r(t, a)T (t, a) = ρ(t, a ′ , a)r(t, a ′ )T (t, a ′ ),

4) r(t, a)T (t, a)r(t, a ′ )T (t, a ′ ) = 0 ⇒ ρ(t, a, a ′ ) = 0.

Properties 1) and 2) follow from the fact that ρ(t, a, a ′ ) is a proportion so that it is always between zero and one, and its total sum equals one. Property 3) comes from the fact that the total number of pairs of individuals of age a with individuals of age a ′ needs to be equal to the total number of pairs of individuals of age a ′ with individuals of age a. Moreover, if there are no active individuals, then there is no pair formation, which leads to property 4).

13.8.2.1

Total Proportionate Mixing

Suppose that the pair formation is a separable function such that ρ(t, a, a ′ ) = ρ1 (t, a)ρ2 (t, a ′ ).

(13.43)

446

13 Disease Transmission Models with Age Structure

It follows from property 2 that 

∞ 0

ρ(t, a, a ′ )da ′ =



∞ 0

ρ1 (t, a)ρ2 (t, a ′ )da ′ = ρ1 (t, a)



∞ 0

ρ2 (t, a ′ )da ′ = 1,

for all t. Hence ρ1 (t, a) =  ∞ 0

1 ρ2 (t, a ′ )da ′

is independent of a. Denote it by L(t). Then ρ(t, a, a ′ ) = L(t)ρ2 (t, a ′ ).

(13.44)

It follows from property 3) and (13.44) that L(t)ρ2 (t, a ′ )r(t, a)T (t, a) = ρ(t, a ′ , a)r(t, a ′ )T (t, a ′ ).

(13.45)

Integrating (13.45) with respect to a from 0 to ∞ yields L(t)ρ2 (t, a ′ )



∞ 0

r(t, a)T (t, a)da = r(t, a ′ )T (t, a ′ ).

(13.46)

Hence r(t, a ′ )T (t, a ′ ) L(t)ρ2 (t, a ′ ) =  ∞ , 0 r(t, a)T (t, a)da

(13.47)

which implies that ρ(t, a, a ′ ) satisfies

r(t, a ′ )T (t, a ′ ) . ρ(t, a, a ′ ) =  ∞ 0 r(t, a)T (t, a)da

(13.48)

Notice that the right-hand side in (13.48) is the fraction of the total partners of age a ′ in the population, or the availability of partners of age a ′ . A pair formation or mixing function satisfying (13.45) is called a total proportionate mixing. Such a mixing depends completely on the availability of partners and is a kind of random mixing. While it may be appropriate to assume a proportionate mixing or random mixing in special cases such as modeling of HIV/AIDS for homosexual men, in general, it is necessary to assume the pair-formation or mixing function to be nonseparable. It is possible to show that proportionate mixing is the only separable pair formation [8].

13.9 A Multi-Age-Group Malaria Model

447

13.9 A Multi-Age-Group Malaria Model Consider a human population in which malaria spreads. Divide the human population into four classes: susceptible, exposed, infective, and recovered who are recovered and also immune from re-infection. Denote the densities by s(a, t), e(a, t), i(a, t), and r(a, t), respectively. We further divide the human population into n age groups with group i consisting of ages a ∈ [ai−1 , ai ), such that the number of individuals in the ith group the respective epidemiological classes  ain  ai  ai i i(a, t)da, and s(a, t)da, Ei (t) = ai−1 e(a, t)da, Ii (t) = ai−1 are Si (t) = ai−1  ai Ri (t) = ai−1 r(a, t)da for i = 1, 2, · · · , n. Then, based on the similar derivation in Sect. 13.6, we obtain the following system of ordinary differential equations for the disease transmission in humans: S1′ (t) = B(t) − (μ1 + c1 )S1 − λ1 (t)S1 ,

Sj′ (t) = cj −1 Sj −1 − λj (t)Sj − (μj + cj )Sj ,

j = 2, . . . , n,

E1′ (t) = λ1 (t)S1 − (μ1 + ǫ1 + c1 )E1 ,

Ej′ (t) = λj (t)Sj + cj −1 Ej −1 − (μj + ǫj + cj )Ej ,

j = 2, . . . , n,

I1′ (t) = ǫ1 E1 − (μ1 + γ1 + ω1 + c1 )I1 ,

Ij′ (t) = ǫj Ej + cj −1 Ij −1 − (μj + γj + ωj + cj )Ij ,

j = 2, . . . , n,

R1′ (t) = γ1 I1 − (μ1 + c1 )R1 ,

Rj′ (t) = cj −1 Rj −1 + γj Ij − (μj + cj )Rj ,

j = 2, . . . , n,

(13.49)

where B(t) is the recruitment rate of the susceptible class, μi the age specific natural death rate, ωi the age specific disease induced death rates, ηi the age progression rate, ǫi the age specific disease progression rate, and γi the age specific recovery rate. The infection rates λj (t) for humans are related to the vector (mosquito) population and are given by λj (t) =

bβj Iv (t) bNv (t) Iv (t) βj = , N (t) Nv (t) N (t)

j = 1, . . . , n,

(13.50)

where b is the number of bites on per mosquito in unit time, Nv the

humans n total mosquito population, N = j =1 (Sj + Ej + Ij + Rj ) the total size of human population, Iv the number of infective mosquitoes, and βj the probability of infection per bite for group i.

448

13 Disease Transmission Models with Age Structure

Let Sv and Ev denote the numbers of susceptible and exposed mosquitoes, and Nv = Sv + Ev + Iv . The dynamics of the mosquito population are described by the equations Sv′ (t) = Mv − λv Sv − μv Sv ,

Ev′ (t) = λv (Nv − Ev − Iv ) − (μv + ǫv ),

(13.51)

Iv′ (t) = ǫv Ev − μv Iv ,

where Mv is an input flow of susceptible mosquitoes, μv is the natural death rate of mosquitoes, ǫv is the disease progression rate for exposed mosquitoes, and λv is the infection rate for mosquitoes given by λv (t) = b

 n  βvj Ij (t) j =1

N (t)

.

(13.52)

Here βvj are the infection rate of mosquitoes from biting infected humans in group j . Readers are referred to [30] for model analysis.

13.10 *Project: Another Malaria Model In this project, we consider an SIS system for the human host and an SI system for the mosquitoes. Let s(a, t) and i(a, t) denote the densities for susceptible and infective hosts, and let n(a, t) = s(a, t) + i(a, t). The system of PDEs for hosts reads st (a, t) + sa (a, t) = −λ(a, t)s(a, t) − μ(a)s(a, t) + γ (a)i(a, t), it (a, t) + ia (a, t) = λ(a, t)s(a, t) − (γ (a) + μ(a))i(a, t),

(13.53)

where λ(a, t) = bβ(a)

Iv (t) N (t)

with boundary and initial conditions s(0, t) =



0

∞

ρ(a)n(a, t)da, i(0, t) = 0, s(a, 0) = s0 (a), i(a, 0) = i0 (a).

13.10 *Project: Another Malaria Model

449

The function ρ(a) denotes the age dependent birth rate. The equations for mosquitoes are Sv′ (t) = Mv − λv (t)Sv (t) − μv Sv (t), Iv′ (t) = λv (t)Sv (t) − μv Iv (t),

(13.54)

where λv (t) = b



∞

βv (a)

0

i(a, t) da, N(t)

with initial conditions Sv (0) = Sv0 ≥ 0, Iv (0) = Iv0 ≥ 0. All other parameters have the same meaning as in (13.51). For the questions in this project, assume that the host population has reached the stable age distribution with constant total population size, i.e.,  ∞  ∞ a − 0 μ(u)du ˜ n(a) ˜ = βπ(a), π(a) = e , N= n(a)da ˜ =β π(a)da, 0

0

∞

˜ t)da = 1 holds. where β is a positive constant and the constraint 0 ρ(a)n(a, Assume also that the total mosquito population has reached the steady state N˜ v = Mv /μv . Question 1 (a) Show that the basic reproduction number has the form  ∞ I (a) ˜ R0 = bβv (a) Nv da, N˜ 0

(13.55)

where I (a) =



a

bβ(σ )

0

βπ(σ ) 1 −  a (γ (τ )+μ(τ ))dτ dσ. e σ N˜ μv

(13.56)

(b) Provide the biological interpretation for the quantities I (a) and R0 . Question 2 Show that the disease-free equilibrium of the system (13.53)–(13.54) is locally asymptotically stable if R0 < 1, and it is unstable if R0 > 1. Question 3 Assume that all age-dependent parameters are constants in the age intervals [ak−1 , ak ), k = 1, · · · , m, i.e., ρ(a) = ρk , β(a) = β, γ (a) = γk , μ(a) = ˜ Let μk . Let n(t, a) = n(a) = βπ(a) with constant total population sizes N. Sk (t) =



ak

s(a, t)da,

ak−1

λk (t) = bβ

Iv (t) , N˜

Ik (t) =

λv (t) = b

m k=1



ak

i(a, t)da, ak−1

βv

Ik (t) , N˜

Nk =



ak

ak−1

k = 1, 2, · · · , m.

n(a)da,

450

13 Disease Transmission Models with Age Structure

Then the ODE system corresponding to the system (13.53)–(13.54) is S1′ (t) = Λ − (α1 + λ1 (t) + μ1 )S1 (t) + γ1 I1 (t),

Sk′ (t) = −(αk + λk + μk )Sk + γk Ik + αk−1 Sk−1 (t), k = 2, . . . , m, I1′ (t) = λ1 (t)S1 (t) − (γ1 + μ1 + α1 )I1 (t),

(13.57)

Ik′ (t) = λk Sk − (γk + μk + αk )Ik + αk−1 Ik−1 (t), k = 2, . . . , m, Iv′ (t) = λv (t)(N˜ v − Iv ) − μv Iv ,

where αk = π(ak )/

 ak

ak−1

π(a)da, k = 1, . . . , m − 1, and αm = 0.

(a) Show that the disease-free equilibrium (DFE), E0 , of system (13.57) is given by E0 =

5

7 6 6 β k−1 β n−1 β j =1 αj j =1 αj , . . . , 6k , 0, . . . , 0, N¯ v , 0 . , . . . , 6n α + μ1 j =1 (αj + μj ) j =1 (αj + μj )

(13.58)

Note that the first m components of E0 describe the age distribution of the population at the DFE, i.e., β

Nk = 6k

6k−1

j =1 αj

j =1 (αj

+ μj )

,

k = 1, 2, · · · , m

(αm = 0).

(13.59)

(b) Derive the basic reproduction number R˜ 0 for the ODE system (13.57). (c) Show that E0 is locally asymptotically stable if R˜ 0 < 1, and unstable if R˜ 0 > 1. Question 4 Verify the result in part (c) of Question 3 by numerical simulations. Consider the case of m = 2 age groups (e.g., children and adults), and assume that R˜ 0 can be varied by changing the value of β. To adopt different ratio of children to adults, the value of μ2 can be determined by the ratio N1 /N2 . One set of possible parameter values can be found in Table 13.1. (a) Choose the values and βk , k = 1, 2, such that R˜ 0 < 1 and plot I1 (t) and I2 (t) vs. time. (b) Repeat part (a) but use different values of βk for which R˜ 0 > 1. (c) Assume that drug treatment can increase the recovery rate γk . Plot R˜ 0 as a function of γ1 or γ2 or both. (d) Consider various scenarios based on treatment in children or adults alone or both. Plot I1 (t) and I2 (t) vs. time and observe how they change with γk . References: [14, 17, 29, 33].

13.11 Project: A Model Without Vaccination

451

Table 13.1 Parameter values and ranges for numerical simulations Parameters Fixed values Mv μv β Values with ranges b

Two age groups

References

150 0.03 0.45N (μ1 + α1 )

[17, 29]

0.2 (0.1, 0.25)

Age-dependent μk αk βk βvk γk

Age group 1 0.0002 1/15/365 0.03 0.3 1/100

[14, 17] Age group 2 α1 N¯ 1 /N¯ 2 0 0.015 0.24 1/80

Reference [33] [33] [14] [14] [14]

13.11 Project: A Model Without Vaccination Consider the model in the previous section but without vaccination, i.e., st (t, a) + sa (t, a) lt (t, a) + la (t, a) it (t, a) + ia (t, a) jt (t, a) + ja (t, a)

= −β(a)c(a)B(t)s(t, a) − μ(a)s(t, a), = β(a)c(a)B(t)(s(t, a) + σj (t, a)) − (k + μ(a))l(t, a), = kl(t, a) − (r + μ(a))i(t, a), = ri(t, a) − σβ(a)c(a)B(t)j (t, a) − μ(a)j (t, a), (13.60)

where B(t) =



∞

0

i(t, a ′ ) p(t, a ′ )da ′ , n(t, a ′ )

with initial and boundary conditions: s(t, 0) = Λ, l(t, 0) = i(t, 0) = j (t, 0) = 0, s(0, a) = s0 (a), l(0, a) = l0 (a), i(0, a) = i0 (a), j (0, a) = j0 (a). The function p is the same as defined in the previous section. Assume that the total population is at the stable age distribution, i.e., n(t, a) = ΛF (a) =: N (a) and let ∞ p(t, a) = p∞ (a) = c(a)N(a)/ 0 c(b)N(b)db. (a) Introduce the fractions u(t, a) =

l(t, a) i(t, a) j (t, a) s(t, a) , x(t, a) = , y(t, a) = , z(t, a) = . n(t, a) n(t, a) n(t, a) n(t, a)

Derive a system for the fractions u, x, y, and z.

452

13 Disease Transmission Models with Age Structure

(b) Derive a formula for the basic reproduction number R0 (c) Show that the DFE, (u(a), x(a), y(a), z(a)) = (1, 0, 0, 0), is locally asymptotically stable when R0 < 1 and unstable when R0 > 1. (d) An endemic steady state of the system is a time-independent solution (u∗ (a), x ∗ (a), y ∗ (a), z∗ (a)),

with x ∗ > 0.

Find an endemic steady state.

13.12 Project: An SI S Model with Age of Infection Structure Consider the following system which describes an SIS model with infectiousness dependent on age-of-infection a: ⎧ ∞ ⎪ S ′ (t) = μN − [μ + λ(t)] S(t) + 0 δγ (a)i(t, a) da ⎪ ⎪ ⎪ ⎪ ⎨ it (t, a) + ia (t, a) = −[μ + δγ (a)]i(t, a) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ i(0, t) = λ(t)S(t), S(0) = S0 > 0, i(0, a) = i0 (a) ≥ 0

(13.61)

where

λ(t) =

β N



∞

p(a)i(t, a)da

0

and N(t) = S(t) + I (t),

I (t) =



∞

i(t, a)da.

0

The function p(a) ∈ (0, 1) describes the fraction of infected individuals who are infectious at infection age a, β is the transmission coefficient for infectious individuals, γ (a) is the age-dependent recovery rate, and δ is a factor representing the increased recovery rate due to treatment. (a) Denote the incidence function by v(t) = λ(t)S(t). Solve the i equation using the method of characteristics and express the solution as a function of v. (b) Derive a system of integral equations using variables N (t) and v(t) that is equivalent to the system (13.61). (c) Derive a formula for the reproduction number R0 . (d) Show that an endemic steady state E exists when R0 > 1, and determine how the disease level at the steady-state E depends on R0 . (e) Examine the dependence of E and R0 on the treatment effort δ.

References

453

13.13 Exercises 1. Consider the model (13.33) in Sect. 13.8.1. (a) Show that lim n(t, a) = ΛF (a)

t→∞

c(a)F (a) lim p(t, a) =  ∞ := p∞ (a), 0 c(b)F (b)db

t→∞

where

F (a) = e−

a 0

μ(s)ds

.

(b) Show that at the disease-free steady state E0 s(a) v(a) = Fψ (a), = 1 − Fψ (a), n(a) n(a) l(a) = i(a) = j (a) = 0, n(a) = ΛF (a),

(13.62)

where Fψ (a) = e−

a 0

ψ(b)db

.

(c) Let R0 denote the basic reproduction number. Show that R0 =



0

∞ ∞ 0

 k p∞ (α + τ )β(α)c(α) e−kτ − e−rτ dτ dα. r −k

References 1. Anderson, R.M. & R.M. May (1991) Infectious Diseases of Humans. Oxford University Press (1991) 2. Anderson, R.M. & R.M. May (1979) Population biology of infectious diseases I, Nature 280: 361–367. 3. Anderson, R.M., and R.M. May (1984) Spatial, temporal, and genetic heterogeneity in host populations and the design of immunization programmes, Math. Med. Biol. 1(3), 233–266. 4. Andreasen, V. (1995) Instability in an SIR-model with age dependent susceptibility, Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. 1, Theory of Epidemics (O. Arino, D. Axelrod, M. Kimmel, M. Langlais, eds.), Wuerz, Winnipeg: 3–14. 5. Blythe, S.P. and C. Castillo-Chavez (1989) Like-with-like preference and sexual mixing models, Math. Biosci. 96: 221–238. 6. Blythe, S.P., C. Castillo-Chavez, J. Palmer & M. Cheng (1991) Towards a unified theory of mixing and pair formation, Math. Biosc. 107: 379–405.

454

13 Disease Transmission Models with Age Structure

7. Busenberg, S. and C. Castillo-Chavez (1989) Interaction, pair formation and force of infection terms in sexually transmitted diseases, in: Mathematical and Statistical Approaches to AIDS Epidemiology, (C. Castillo-Chavez, ed.), Lecture Notes in Biomathematics, Vol. 83, SpringerVerlag, New York, (1989), pp. 289–300. 8. Busenberg, S. and C. Castillo-Chavez (1991) A general solution of the problem of mixing of subpopulations and its application to risk- and age-structured epidemic models for the spread of AIDS, IMA J. Math. Appl. Med. Biol. 8: 1–29. 9. Capasso, V. (1993) Mathematical Structures of Epidemic Systems, Lect. Notes in Biomath. 83, Springer-Verlag, Berlin-Heidelberg-New York. 10. Castillo-Chavez, C. and Z. Feng (1998) Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosci. 151(2), 135–154. 11. Castillo-Chavez, C., H.W. Hethcote, V. Andreasen, S.A. Levin, and W.M. Liu (1989) Epidemiological models with age structure, proportionate mixing, and cross-immunity, J. Math. Biol. 27(3), 233–258. 12. Castillo-Chavez, C. and S. P. Blythe (1989) Mixing Framework for Social/Sexual Behavior, in: Mathematical and Statistical Approaches to AIDS Epidemiology, (C. Castillo-Chavez, ed.), Lecture Notes in Biomathematics 83, Springer-Verlag, New York, pp. 275–288. 13. Cha, Y., M. Ianelli, & F. Milner (1998) Existence and uniqueness of endemic states for the age-structured S-I-R epidemic model, Math. Biosc. 150: 177–190. 14. Chitnis, N., J.M. Hyman, and J.M. Cushing (2008) Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol 70:1272–1296. 15. Dietz, K. (1975) Transmission and control of arbovirus diseases, Epidemiology (D. Ludwig, K. Cooke, eds.), SIAM, Philadelphia: 104–121. 16. Dietz, K., and D. Schenzle (1985) Proportionate mixing models for age-dependent infection transmission, J. Math. Biol. 22(1), 117–120. 17. Garret-Jones, C. and G.R. Shidrawi (1969) Malaria vectorial capacity of a population of Anopheles gambiae. Bull. Wld Hlth Org. 40:531–545. 18. Gurtin, M.L. and R.C. MacCamy (1974) Nonlinear age dependent population dynamics, Arch. Rat. Mech. Analysis, 54: 281–300. 19. Hadeler, K.P., R. Waldstatter, and A. Worz-Busekros (1988) Models for pair-formation in bisexual populations, J. Math. Biol. 26: 635–649. 20. Hethcote, H.W. (2000) The mathematics of infectious diseases, SIAM Review 42: 599–653. 21. Hoppensteadt, F.C. (1975) Mathematical Theories of Populations: Demographics, Genetics and Epidemics, SIAM, Philadelphia. 22. Iannelli, M. (1995) Mathematical Theory of Age-Structured Population Dynamics, Appl. Mathematical Monographs, C.N.R.. 23. Kot, M. (2001) Elements of Mathematical Ecology. Cambridge University Press. 24. May, R.M. (1986) Population biology of macroparasitic infections, in Mathematical Ecology; An Introduction, (Hallam, T.G., Levin, S.A. eds.), Biomathematics 18, Springer-Verlag, BerlinHeidelberg-New York: 405–442. 25. May, R.M. & Anderson, R.M. (1979) Population biology of infectious diseases II, Nature 280: 455–461. 26. McKendrick, A.G. (1926) Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc. 44: 98–130. 27. Müller, J. (1998) Optimal vaccination patterns in age structured populations, SIAM J. Appl. Math. 59: 222–241. 28. Murray, J.D. (2002) Mathematical Biology, Vol. I, Springer-Verlag, Berlin-Heidelberg-New York. 29. Paaijmans, K. Weather, water and malaria mosquito larvae, 2008. 30. Park, T. (1984) Age-Dependence in Epidemic Models of Vector-Borne Infections, Ph.D. Dissertation, University of Alabama in Huntsville, 2004. 31. Sharpe, F.R., and A.J. Lotka (1911) A problem in age-distribution. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 21(124): 435–438.

References

455

32. Thieme, H.R. (2003) Mathematics in Population Biology, Princeton University Press, Princeton, N.J. 33. United nations, department of economic and social affairs, population division, population estimates and projections sections. Retrieved January, 2013 from http://esa.un.org/wpp/index. htm. 34. Waltman, P. (1974) Deterministic Threshold Models in the Theory of Epidemics, Lect. Notes in Biomath. 1, Springer-Verlag, Berlin-Heidelberg-New York. 35. Webb, G.F. (1985) Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, New York.

Chapter 14

Spatial Structure in Disease Transmission Models

14.1 Spatial Structure I: Patch Models With the advent of substantial intercontinental air travel, it is possible for diseases to move from one location to a completely separate location very rapidly. This was an essential aspect of modeling SARS during the epidemic of 2002–2003, and has become a very important part of the study of the spread of epidemics. Mathematically, it has led to the study of metapopulation models or models with patchy environments and movement between patches [4–7, 33, 38]. These models, which are the focus of this section, are called metapopulation models. They usually consist of a system (often a large system) of ordinary differential equations with some coupling between patches. A patch can be a city, community, or some other geographical region. In real life, a metapopulation model for the spread of a communicable disease should consider all the locations for which there are interactions. An example would be two distant cities with some air travel between them but no contact otherwise. It is possible that not all patches are linked directly. For example, we might think of a system consisting of a city and two suburbs, with contact between occupants of each suburb and the city but not between occupants of the two suburbs. Thus a model must keep track of both the patches and the links between them, and should be described in terms of a graph. In the interest of simplicity, we will confine our attention to models consisting of only two patches, but it is important to be aware of the complications of scale. A more thorough description may be found in [3].

14.1.1 Spatial Heterogeneity Consider a basic SI R compartmental model. We divide the population into two connected sub-populations. Let Si , Ii , Ri denote, respectively, the number of © Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_14

457

458

14 Spatial Structure in Disease Transmission Models

susceptible, infective, and recovered individuals in Patch i for i = 1, 2. The total population of Patch i is Ni = Si + Ii + Ri . The birth and natural death rate constant μ is assumed to be the same in each patch, so that the total population of each patch remains constant. The average infective period 1/γ is assumed to be the same in each patch. This spatial model can be written for i = 1, 2 as in [24] Si′ = μNi − μSi − λi Si Ii′ = λi Si − (μ + γ )Ii

(14.1)

Ri′ = γ Ii − μRi ,

with the force of infection in Patch i given by a mass action type of incidence λi = βi1 I1 + βi2 I2 . Thus, infective individuals in one patch can infect susceptible individuals in another patch, but there is no explicit movement of individuals in this model. For the SIR model (14.1) the disease-free equilibrium is Si = Ni , Ii = Ri = 0. Using the next generation matrix [39] R0 can be calculated from (14.1) as R0 = ρ(F V −1 ), where the (i, j ) of F V −1 is βij Ni /(d + γ ). For the case that each patch has the same population (i.e., Ni = N) and βij are such that the endemic equilibrium values of Si , Ii , and λi are independent of i, then the endemic equilibrium is given explicitly for R0 > 1 by Si∞ = S∞ =

N , R0

Ii∞ = I∞ =

μN μ+γ



1−

1 R0



,

Ri∞ = R∞ =

γ I∞ , d (14.2)

with λi∞ = λ∞ = μ(R0 − 1). As an example of a symmetric situation that satisfies the above requirements, assume that βij = β if i = j and βij = pβ with p < 1 if i = j . Then the contact rate is the same within each patch and has a smaller value between patches. We may calculate that R0 =

βN (p + 1) , (μ + γ )

which depends on the coupling strength p. We should note that the assumption in the above example that the two patches have the same total population size is quite unrealistic in practice; it is given only as a simple example. If we linearize about the endemic equilibrium and solve the linear approximation, we find that the solutions are damped oscillations about the equilibrium which are phase-locked except for very small values of p. Simulations suggest that, as has been found in other metapopulation models, with larger p values (i.e., stronger between patch coupling) the system effectively behaves much like a single patch.

14.1 Spatial Structure I: Patch Models

459

14.1.2 Patch Models with Travel Sattenspiel and Dietz [33] introduced a metapopulation epidemic model in which individuals are labeled with their city of residence as well as the city in which they are present at a given time. To formulate a demographic model with travel for two patches, let Nij (t) be the number of residents of Patch i who are present in Patch j at a time t. Residents of Patch i leave this patch at a per-capita rate gi ≥ 0 per unit time. For a model with more than two patches we would also need to count the fraction of these travelers going to each patch. Residents of Patch i who are in Patch j return home to Patch i with a per-capita rate of rij ≥ 0 with r11 = r22 = 0. It is natural to assume that gi > 0 if and only if rij > 0. These travel rates determine a directed graph with patches as vertices and edges connecting vertices if the travel rates between them are positive. It is assumed that the travel rates are such that this directed graph is strongly connected. Assume that births occur in the home patch at a per-capita rate μ > 0, and that natural deaths occur in each patch with this same rate. Then the population numbers satisfy the equations Nii′ Nij′

=

2 k=1

2   rik Nik − gi Nii + μ Nik − Nii k=1

= gi Nii − rij Nij − μNij ,

(14.3)

i, j = 1, 2, i = j.

These equations describe the evolution of the number of residents in Patch i who are currently in Patch i and those who are currently in Patch j = i. In the first equation of (14.3) the term μNik represents births in Patch i to residents of Patch i currently in Patch k. The number of residents of Patch i, namely Nir = Ni1 + Ni2 is constant, as is the total population of the two-patch system. With initial conditions Nij (0) > 0, the system (14.3) has an asymptotically stable equilibrium Nˆ ij . We now formulate an epidemic model in each of the patches, with Sij (t) and Iij (t) denoting the number of susceptible and infective individuals resident in Patch i who are present in Patch j at time t. The equations for the evolution of the number of susceptibles and infectives residents in Patch i (with i = 1, 2) are 5 2 7 S I ii ki Nik − Sii κi βiki rik Sik − gi Sii − Sii′ = p +μ Ni k=1 k=1 k=1 2

Iii′

2

2

2

Sii Iki = rik Iik − gi Iii + κi βiki p − (γ + μ)Iii , Ni k=1 k=1

(14.4)

460

14 Spatial Structure in Disease Transmission Models

and for j = i Sij′ = gi Sii − rij Sij − Iij′

2

κj βikj

k=1

Sij Ikj p − μSij Nj

2

Sij Ikj = gi Iii − rij Iij + κj βikj p − (γ + μ)Iij , Nj k=1

(14.5) i, j = 1, 2

p

with Ni = N1i + N2i denoting the number present in Patch i. Here βikj > 0 is the proportion of adequate contacts in Patch j between a susceptible from Patch i and an infective from Patch k that results in disease transmission, βikj > 0 is the average number of such contacts in Patch j per unit time, and γ > 0 is the recovery rate of infectives (assumed the same in each patch). Note that the disease is assumed to be sufficiently mild so that it does not cause death and does not inhibit travel, and it is assumed that individuals do not change disease status during travel. Equations (14.4) and (14.5) together with non-negative initial conditions constitute the SI R metapopulation model. The disease-free equilibrium is given by Sij = Nˆ ij , Iij = 0 for i, j = 1, 2. If the system is at an equilibrium and one patch is at the disease-free equilibrium, then all patches are at the disease-free equilibrium; whereas if one patch is at an endemic disease level, then all patches are at an endemic level. These results hold based on the assumption that the directed graph determined by the travel rates is strongly connected. If this is not the case, then the results apply to patches within a strongly connected component. We may calculate the basic reproduction number R0 for the model (14.4), (14.5) using the next generation matrix approach [14, 39]. As the result is somewhat complicated, we do not give it explicitly here, but we note that R0 depends on the travel rates as well as the epidemic parameters. If R0 < 1, then the disease-free equilibrium is locally asymptotically stable; whereas if R0 > 1, then it is unstable. If the disease transmission coefficients are equal for all populations present in a patch, i.e., βij k = βk for i, j = 1, 2 it is possible to obtain the bounds min Roi ≤ R0 ≤ max Roi ,

i=1,2

i=1,2

(14.6)

where Roi = κi βi /(d + γ ) is the basic reproduction number of Patch i in isolation. Thus if Roi < 1 for all i, the disease dies out; whereas if Roi > 1 for all i, then the disease-free equilibrium is unstable. A change in travel rates g1 , g2 can induce a bifurcation from R0 < 1 to R0 > 1 or vice versa, see [5, Fig. 3a]. Thus travel can stabilize (small travel rates) or destabilize (larger travel rates) the disease-free equilibrium. Numerical simulations support the claim that for R0 > 1, the endemic equilibrium is unique and that R0 acts as a sharp threshold between extinction and invasion of the disease.

14.1 Spatial Structure I: Patch Models

461

Sattenspiel and Dietz [33] have suggested an application of their metapopulation SIR model to the spread of measles in the 1984 epidemic in Dominica. Travel rates of infants, school-age children, and adults are assumed to be different, thus making the model system highly complex and requiring knowledge of much data for simulation. Sattenspiel and coworkers, see [32] and the references therein, have since used this modeling approach for studying other infectious diseases in the historical archives. The SARS epidemic of 2002–2003 spread rapidly through airline transportation from Asia to North America, and if there is an influenza pandemic in the near future it is likely that it will spread in a similar way. Metapopulation models, perhaps with small numbers of traveling infectives, may be a useful approach to modeling such a spread. Because the airline network is complex and because passenger travel data are difficult to acquire, there are substantial technical problems in the formulation of accurate models. However, it is possible that the qualitative insights that can be obtained from simple metapopulation models may be useful.

14.1.3 Patch Models with Residence Times In this chapter we have been examining patch models with travel rates between patches included explicitly. Another possible perspective, which may be more appropriate in some situations, would be to describe patches with residents who spend a fraction of their time in different patches. For example, the spread of an infectious disease from one village to another through people who visit other patches may be a realistic description. Another interpretation could be to assume that individuals spend some of their time in environments more likely to allow disease transmission. We consider an SIR epidemic model in two patches, one of which has a significantly larger contact rate, with short-term travel between the two patches. The total population resident in each patch is constant. We follow a Lagrangian perspective, that is, we keep track of each individual’s place of residence at all times [9, 12, 16]. This is in contrast to an Eulerian perspective, which describes migration between patches. Thus we consider two patches, with total resident population sizes N1 and N2 , respectively, each population being divided into susceptibles, infectives, and removed members. Si and Ii denote the number of susceptibles and infectives, respectively, who are residents in Patch i, regardless of the patch in which they are present. Residents of Patch i spend a fraction pij of their time in Patch j , with 2 i=1

pi1 + pi2 = 1,

i = 1, 2.

βi is the risk of infection in Patch i, and we assume β1 > β2 .

462

14 Spatial Structure in Disease Transmission Models

Each of the p11 S1 susceptibles from Patch 1 who are present in Patch 1 can be infected by infectives from Patch 1, and infectives from Patch 2 who are present in Patch 1. Similarly, each of the p12 S1 susceptibles present in Patch 2 can be infected by infectives from Patch 1, and by infectives from Patch 2 who are present in Patch 2. The number of infectives from both patches who are present in Patch 1 is p11 I1 (t) + p21 I2 (t) and the total number of individuals present in Patch 1 is p11 N1 + p21 N2 . Thus the density of infected individuals in Patch 1 at time t who can infect only individuals currently in Patch 1 at time t, that is, the effective infective proportion in Patch 1 is given by p11 I1 (t) + p21 I2 (t) . p11 N1 + p21 N2 Thus the rate of new infections of members of Patch 1 in Patch 1 is β1 p11 S1

p11 I1 (t) + p21 I2 (t) . p11 N1 + p21 N2

The rate of new infections of members of Patch 1 in Patch 2 is β2 p12 S1

p12 I1 (t) + p22 I2 (t) . p12 N1 + p22 N2

Then the differential equations for S1 and I1 for an SI R infection are given by    p12 I1 (t) + p22 I2 (t) p11 I1 (t) + p21 I2 (t) − β2 p12 S1 p11 N1 + p21 N2 p12 N1 + p22 N2     p12 I1 (t) + p22 I2 (t) p11 I1 (t) + p21 I2 (t) ′ + β2 p12 S1 − γ I1 . I1 = β1 p11 S1 p11 N1 + p21 N2 p12 N1 + p22 N2 (14.7)

S1′ = −β1 p11 S1



There is a corresponding calculation for the rate of new infections of members of Patch 2 in each patch, and the differential equations for S2 and I2 are given by 

   p11 I1 (t) + p21 I2 (t) p12 I1 (t) + p22 I2 (t) − β2 p22 S2 p11 N1 + p21 N2 p12 N1 + p22 N2     p12 I1 (t) + p22 I2 (t) p11 I1 (t) + p21 I2 (t) + β2 p22 S2 − γ I2 . I2′ = β1 p21 S2 p11 N1 + p21 N2 p12 N1 + p22 N2 (14.8) S2′ = −β1 p21 S2

14.1 Spatial Structure I: Patch Models

463

Using the next generation approach to compute the basic reproduction number [39] we define ⎛

F =⎝

+p21 I2 22 I2 p11 β1 S1 pp1111NI11 +p + p12 β2 S1 pp1212NI11 +p +p22 N2 21 N2 +p21 I2 22 I2 p21 β1 S2 pp1111NI11 +p + p22 β2 S2 pp1212NI11 +p +p22 N2 21 N2

⎞ ⎠

and





V =



γ I1 γ I2



,

then F =



B11 B12 B21 B22



and

V =

γ 0 0γ

,

where B11 = N1 B12 = N1 B12 = N2 B22 = N2

5





5

2 β 2 β p12 p11 1 2 + p11 N1 + p21 N2 p12 N1 + p22 N2

p12 p22 β2 p11 p21 β1 + p11 N1 + p21 N2 p12 N1 + p22 N2 p11 p21 β1 p12 p22 β2 + p11 N1 + p21 N2 p12 N1 + p22 N2 2 β 2 β p22 p21 2 1 + p11 N1 + p21 N2 p12 N1 + p22 N2

7

 

7

,

, ,

.

The B matrix explicitly captures the secondary infections produced by Patch 1 and Patch 2 individuals in each environment. For example, B12 collects the Patch 1 residents infected by Patch 2 inhabitants in both environments. Finally, the reproduction number is the largest eigenvalue of the matrix F V −1 , this is

R0 (P) =

B11 + B22 −

 2 + 4B B − 2B B + B 2 B11 12 21 11 22 22 2γ

.

Figure 14.1 shows the effect of mobility on R0 (P) as residence times vary. In the next chapter we show the applications of this approach in the context of Ebola, tuberculosis, and Zika. In the special case of no movement between patches p11 = p22 = 1,

p12 = p21 = 0,

we obtain R0 = max



β1 β2 , γ γ



.

464

14 Spatial Structure in Disease Transmission Models

Fig. 14.1 Effect of mobility in the global R0 . Parameters β1 = 0.09, β2 = 0.02, and γ = 0.05

Integration of the equations for (Si + Ii ) (i = 1, 2) gives γ



∞ 0

Ii (t)dt = Ni − Si (∞),

i = 1, 2

and these relations combined with the result of integrating the equations for Si′ /Si (i = 1, 2) give the final size relations, whose form is quite complicated. Choosing different values for pij gives a way to estimate the effect on the epidemic size of imposing travel restrictions between patches.

14.2 Spatial Structure II: Continuously Distributed Models In the preceding section, we have discussed the spread of a communicable disease from one patch to another. In this section we will discuss the spatial spread of a disease in a single patch because of the (continuous) motion of individuals. The mathematical analysis is based on partial differential equations of reaction– diffusion type. It is technically complicated and requires substantial mathematical background. As references for some of the mathematical details, we suggest [10, Chapter 5], [15, Chapters 9–11], [23, Chapters 15–18], [27, Chapters 11 and 13], [28, Chapters 1 and 2]. An introduction to models for the spatial spread of epidemics may be found in other references such as [2, 11, 14, 19, 37]. One characteristic feature of such models is the appearance of traveling waves, which have been observed frequently in the spread of epidemics through Europe from medieval times to the more recent studies of fox rabies [1, 22, 25, 29]. The asymptotic speed of spread of disease is the minimum wave speed [8, 13, 26, 31, 35, 41]. Models describing spatial spread and including age of infection are analyzed in [17, 18, 20, 28].

14.2 Spatial Structure II: Continuously Distributed Models

465

14.2.1 The Diffusion Equation Let us begin by considering the motion of particles. Here, by a particle we might mean an individual cell, a member of a population, or any object of a set in whose spatial distribution as a function of time we are interested. Our approach is to take a small region of space and to form a balance equation which says that the rate of change of the number of particles in the region is equal to the rate at which particles flow out of the region minus the rate at which particles flow into the region plus the rate of creation of particles in the region. We shall confine ourselves mainly to the case in which the dependence is with respect to a single space coordinate. Let us think of a tube of constant cross section area A and let x denote the distance along the tube from some arbitrary starting point x = 0. We assume that the tube is a bounded region described by the inequalities 0 ≤ x ≤ L. Let u(x, t) be the concentration of particles (number per unit volume) at location x at time t, meaning that in the portion of the tube between x and x +h, with volume Ah the number of particles is approximately Ahu(x, t). By “approximately” we mean that if h is small, the error in this approximation Ahu(x, t) is smaller than a constant multiple of h2 . We let J (x, t) be the flux of particles at location x at time t, by which we mean the time rate of the number of particles crossing a unit area in the positive direction. For every x0 the net rate of flow into the region between x0 and x0 +h is AJ (x0 , t)− AJ (x0 + h, t). We let Q(x0 , t, u) be the net growth rate per unit length at location x0 at time t, representing births and deaths. We have a balance relation on the interval x0 ≤ x ≤ x0 + h, expressing the fact that the rate of change of population size at time t in this interval is equal to the growth rate of population in this interval plus the net flux, and this leads to the conservation law ut (x, t) = Q(x, t, u) −

∂J (x, t). ∂x

(14.9)

In order to obtain a model which describes the population density u(x, t) we must make some assumption which relates the rate of change of flux density ∂J ∂x and the population density u(x, t). If the motion is random, then Fick’s law says that the flux due to random motion is approximately proportional to the rate of change of particle concentration, that is, that J is proportional to ux . If population density decreases as x increases (ux < 0) we would expect J > 0, so that J and ux have opposite sign and thus that J = −Dux with D a constant called the diffusivity or diffusion coefficient. More generally, D could be a function of the location x but we shall confine our attention to constant diffusivity. Equation (14.9) then becomes a second-order partial differential equation

466

14 Spatial Structure in Disease Transmission Models

ut (x, t) = Q(x, t, u) + Duxx (x, t).

(14.10)

Equation (14.10) is a reaction–diffusion equation. If Q = 0, it is called the heat or diffusion equation. It is possible to solve the heat equation explicitly; the solution for −∞ < x < ∞, with u(x, 0) = f (x) is u(x, t) = √

1 4π Dt



0 ≤ t < ∞,

∞

e−

(x−ξ )2 4Dt

f (ξ ) dξ.

−∞

In population ecology, we can translate Fick’s law of diffusion into the statement that the individuals move from a region of high concentration to a region of low concentration in search for limited resources. We must, however, use this law with caution when modeling spatial spread of infectious diseases since the individual movement behaviors may be altered during the course of outbreaks of diseases. For Eq. (14.10) to have a unique solution, we need to impose additional conditions. It is possible to establish the following result. The diffusion equation (14.10) with a specified initial condition u(x, 0) = f (x) for 0 ≤ x ≤ L and boundary conditions for x = 0 and x = L has a unique solution for 0 ≤ x ≤ L, 0 ≤ t < ∞. The boundary conditions may specify the value of u or the value of ux for x = 0 and x = L. Such a problem is called an initial boundary value problem. We could also consider problems in an infinite tube defined by −∞ < x < ∞ for which no boundary conditions are required, or a semi-infinite tube 0 ≤ x < ∞ for which a boundary condition is required only at x = 0. In each case there is a unique bounded solution for any specified initial condition u(x, 0) = f (x) for −∞ ≤ x < ∞ or u(x, 0) = f (x) for 0 ≤ x < ∞. A boundary condition specifying that the solution must vanish at a boundary (called an absorbing boundary) may be taken to say that an individual leaving the region must die immediately. This is an idealization, but we may think of a large region with u = 0 far enough away. A boundary condition specifying that ux must vanish at the boundary may be taken to say that the population is confined to the region and there is no flow across the boundary. There are several types of initial condition which may arise. One possibility is that particles are absent initially, u(x, 0) = 0 and enter through the boundary. A second possibility is that particles are inserted at a single point x0 , u(x, 0) = u0 δ(x − x0 ). Here, δ(x) denotes the delta “function,” which is zero except for x = 0 and  ∞ δ(x) dx = 1 (14.11) −∞

and if f is continuous, then

14.2 Spatial Structure II: Continuously Distributed Models



∞ −∞

f (x)δ(x − a) dx = f (a).

467

(14.12)

A third kind of initial condition would be to specify a constant initial concentration, u(x, 0) = u0 for 0 ≤ x ≤ L.

14.2.2 Nonlinear Reaction–Diffusion Equations In this section we consider reaction–diffusion equations containing a nonlinear growth rate g(u) with g(0) = g(K) = 0, g ′ (u) > 0 for 0 ≤ u < K and g ′ (K) < 0. Thus, we shall consider the equation ut (x, t) = Duxx (x, t) + g(u).

(14.13)

This equation could describe a population with diffusion in space, and births and deaths given by the function g(u). We begin by looking for solutions u(t) which are independent of x. Then uxx = 0 and Eq. (14.13) reduces to the ordinary differential equation u′ = g(u).

(14.14)

Note that K is the carrying capacity, every bounded solution of (14.14) approaches the equilibrium K as t → ∞. For solution patterns in space, we can consider time-independent solutions, i.e., ut = 0. In this case, Eq. (14.13) reduces to the following second-order ordinary differential equation: Du′′ + g(u) = 0.

(14.15)

Let v = u′ , then v ′ = −g(u)/D, Eq. (14.15) is equivalent to the following firstorder system:  ′   u v = . v −g(u)/D

(14.16)

From g(0) = g(K) = 0 and g ′ (u) > 0 for 0 ≤ u < K, we know that system (14.16) has equilibria (u∞ , v∞ ) = (0, 0) and (K, 0). The Jacobian matrix is 

0 ′

∞) − g (u D

 1 . 0

(14.17)

468

14 Spatial Structure in Disease Transmission Models

Since g ′ (0) > 0, both eigenvalues at the equilibrium (0, 0) are positive and this equilibrium is asymptotically stable. Since g ′ (K) < 0, there is one positive and one negative eigenvalue and the equilibrium (K, 0) is a saddle point. Nonlinear reaction–diffusion equations may have traveling wave solutions. A traveling wave solution has the form u(x, t) = U (x − ct) for some constant c. In this case, ux (x, t) = U ′ (x − ct), uxx (x, t) = U ′′ (x − ct), and ut (x, t) = (−c)U ′ (x − ct). Thus, from Eq. (14.13), the following equation holds: −cU ′ (x − ct) = g[U (x − ct)] + DU ′′ (x − ct). Let z = x − ct, then the function U (z) must satisfy the second-order ordinary differential equation in z DU ′′ + cU ′ + g(U ) = 0.

(14.18)

The first-order system equivalent to (14.18) is U′ = V,

V′ = −

V g(U ) −c , D D

(14.19)

whose equilibria are (U∞ , V∞ ) = (0, 0) and (K, 0). The Jacobian matrix at (U∞ , 0) is   0 1 . g ′ (U ) − D∞ − Dc From g ′ (K) < 0 we know that (K, 0) is a saddle point. The equilibrium (0, 0) is an asymptotically stable node if c2 > 4Dg ′ (0) and a stable point if c2 < 4Dg ′ (0). By studying the phase portrait of the system it is possible to show that, if (0, 0) is a node, then there is an orbit connecting the saddle point as z → −∞ and the equilibrium at (0, 0) as z → ∞. This orbit corresponds to a wave solution u(x, t) traveling to the right, as shown in Fig. 14.2.

14.2.3 Disease Spread Models with Diffusion If we take the simple endemic SI R model (3.1) considered in chapter 3 (with Λ being a constant and d = 0) and add diffusion, we obtain the reaction–diffusion model: St = Λ − μS − βSI + DSxx It = βSI − (α + d + μ)I + DIxx .

(14.20)

14.2 Spatial Structure II: Continuously Distributed Models

469

Fig. 14.2 A connecting orbit of Eq. (14.13)

A search for traveling wave solutions would lead to a four-dimensional system of ordinary differential equations. This approach can be carried out but is technically complicated, and we will not pursue it. Instead, we will consider a case study in which it is biologically reasonable to assume that susceptible members of the population do not diffuse, such as the spread of rabies in continental Europe during the period 1945–1985. This will permit a search for traveling wave solutions that requires the analysis of only a two-dimensional system. The epidemic began on the edge of the German/Polish border, and its front moved westward at an average speed of about 30–60 km per year. The spread of the epidemic was essentially determined by the ecology of the fox population as foxes are the main carrier of the rabies under consideration. A model was formulated in [22] to describe the front of the wave, its speed, and the total number of foxes infected after the front passes, and the connection of the wave speed to the so-called propagation speed of the disease. We formulate a model describing susceptible (S) and infective (I ) foxes. Assume that susceptible foxes are territorial and do not diffuse, but the rabies virus induces a loss of sense of territory. Consider the case when the population size has been scale to 1, i.e., 0 < S(x, t) ≤ 1 and 0 ≤ I (x, t) < 1. Assume also a uniform initial density for the susceptibles with S0 = 1. The simplest epidemic model under these assumptions is St (x, t) = −βS(x, t)I (x, t) It (x, t) = DIxx (x, t) + βS(x, t)I (x, t) − αI (x, t),

(14.21)

where β is the transmission coefficient, α is the disease death rate of infective foxes, and D is the diffusion coefficient.

470

14 Spatial Structure in Disease Transmission Models

Consider the traveling wave solution with speed c: u(z) = S(x − ct),

v(z) = I (x − ct),

where z = x − ct, and u and v are the waveforms (or wave profiles). Substituting the above special form into the system (14.21), we obtain the system of ordinary differential equations: Dv ′′ + cv ′ + βuv − αv = 0,

cu′ − βuv = 0,

(14.22)

where primes denote differentiation with respect to z. Assume the boundary conditions: u(−∞) = a,

u(∞) = 1,

v(−∞) = v(∞) = 0,

(14.23)

where a is a constant to be determined. Substituting the second equation in (14.22) into the first equation and using the boundary conditions we obtain the system β uv, c  c α v′ = 1 − u − v + ln u . D β

u′ =

(14.24)

Let (u∞ , v∞ ) denote an equilibrium of system (14.24). Then v∞ = 0. For u∞ , it is either 0 or a solution of the equation u−1=

α ln u. β

(14.25)

For 0 < u∞ < 1, a solution of (14.25), denoted by a, exists if and only if a
1, we know that J (E1 ) has negative trace and negative determinant, and hence, E1 = (a, 0) is a saddle point. Let  c∗ = 2 βD(1 − α/β),

then E2 = (1, 0) is a stable node if c > c∗ and a stable focus if c < c∗ . Hence, a traveling wave solution satisfies c > c∗ , in which case there is a connecting orbit from E1 to E2 . In the model considered here we have neglected many important factors, including births and natural deaths and the long latent period fox rabies. More accurate models can predict not only the observed wave pattern but also give a close approximation to the shape of the epidemic wave. Some additional sources of information about rabies modeling are [21, 25, 29, 30, 42]. With diffusion in both S and I , other difficult questions arise. One question is diffusive instability, meaning that an equilibrium is asymptotically stable for the system of ordinary differential equations obtained by a search for solutions that are constant in time but unstable for the system with diffusion. In general, diffusion tends to have a stabilizing effect and diffusive instability requires very specific conditions on the coefficients. We have looked only at diffusion in one-dimensional space. In the extension to higher space dimensions, the term uxx can be replaced by the Laplacian of the function u. In many problems for two-dimensional space there is radial symmetry, which can be incorporated by describing the Laplacian in polar coordinates and assuming u to be independent of the angular coordinate. If the radial variable is denoted by r, the term Duxx would be replaced by urr + ur /r. Diffusion problems may be mathematically very complicated, and they require a considerable amount of mathematical background. One important possibility is the formation of spatial patterns, first suggested by A. M. Turing in 1952 [36]. These require more knowledge of partial differential equations than wish to assume. Some examples of pattern formation in diffusive epidemic models may be found in [34, 40] and further information may be found in [27].

14.3 Project: A Model with Three Patches The epidemic patch model, (14.4) and (14.5), is for the case of two patches. We can consider an extension of the model to include three patches. In this case, individuals leaving Patch i can travel to either one of the two other patches. Let mj i ≥ represent the

3fractions of individuals moving into Patch j from Patch i. Then mii = 0, rii = 0, j =1 mj i = 1, and gi mj i denotes the travel rate entering Patch j from Patch i. Consider the case in which the transmission coefficient βikj depends only on the

472

14 Spatial Structure in Disease Transmission Models

Patch j where the transmission occurs, i.e., βikj = βj . Then the system on resident Patch i (with i = 1, 2, 3) reads Sii′ = Iii′

3 k=1

rik Sik − gi Sii −

3

3

κi βi

k=1

3

3

 Sii Iki + μ Nik − Sii p Ni k=1

(14.27)

Sii Iki = rik Iik − gi Iii + κi βi p − (γ + μ)Iii , Ni k=1 k=1

and for j = i Sij′ = gi mj i Sii − rij Sij − Iij′

3

κj βj

k=1

Sij Ikj p − μSij Nj

3

(14.28)

Sij Ikj = gi mj i Iii − rij Iij + κj βj p − (γ + μ)Iij , Nj k=1

p

with Ni = N1i + N2i + N3i , the number present in Patch i. Question 1 The total population sizes satisfy the following equations: Nii′ = Nij′

3 k=1

3   Nik − Nii , rik Nik − gi Nii + μ

i = 1, 2, 3,

k=1

= gi mj i Nii − rij Nij − μNij ,

(14.29)

i = j.

(a) Show that the total resident population in Patch i, 3k=1 Nik , remains constant

at all time. Let Ni0 = 3k=1 Nik for i = 1, 2, 3. (b) Show that the system (14.29) has the asymptotically stable equilibrium Nˆ ii =

5

1 + gi

and for j = i Nˆ ij =

1

3

mki k=1 μ+rik

gi mj i Nˆ ii . μ + rij

7

Ni0 ,

(14.30)

(14.31)

p Question 2 Let Nˆ iq be as given in (14.30) and (14.31), and let Nˆ q = 3i=1 Nˆ iq . It is easy to show that R0i = κi βi /(μ + γ ) is the isolated basic reproduction number of Patch i (i.e., when there is no travel between patches). Consider the order of infective variables

14.3 Project: A Model with Three Patches



473

 I11 , I12 , I13 , I21 , I22 , I23 , I31 , I32 , I33 .

(a) Show that the basic reproduction number R0 for the model (14.27)–(14.28) is given by the dominant eigenvalue of the matrix F V −1 , where F is a block matrix with nine blocks, and each block Fij is a 3 × 3 matrix with the form Fij = diag(fij q ) with fij q = κq βq

Nˆ iq p, Nˆ q

q = 1, 2, 3

and ⎞ a1 −r12 −r13 0 0 0 0 0 0 ⎜−g1 m21 b12 0 0 0 0 0 0 0 ⎟ ⎟ ⎜ ⎜−g m 0 0 0 0 0 ⎟ ⎟ ⎜ 1 31 0 b13 0 ⎜ 0 0 0 0 ⎟ 0 0 b21 −g2 m12 0 ⎜ ⎟ ⎜ ⎟ V =⎜ 0 a2 −r23 0 0 0 ⎟, 0 0 −r21 ⎜ ⎟ ⎜ 0 0 0 ⎟ 0 0 0 −g2 m32 b23 0 ⎜ ⎟ ⎜ 0 0 0 0 0 0 b31 0 −g3 m13 ⎟ ⎜ ⎟ ⎝ 0 0 0 0 0 0 0 b32 −g3 m23 ⎠ 0 0 0 0 0 0 −r31 −r32 a3 ⎛

where ai = gi + γ + μ and bik = rik + γ + μ. (b) Consider the special case when βi = β for i = 1, 2, 3. Fix all parameters except β. Then R0 = R0 (β) is a function of β. Consider the set of parameters: γ = 1/25, μ = 1/(75 × 365), g1 = 0.01, g2 = 0.02, g3 = 0.03, mij = 0.5 for i = j , rij = 0.05 for i = j , κi = κ = 1 and N0i = N0 = 1500 for i = 1, 2, 3. (i) Plot R0 as a function of β. What is the threshold value βc such that R0 (β) < 1 for all β < βc ? (ii) Figure 14.3 shows the number of susceptible and infective individuals of three resident populations for the case of β = 0.025 with different values of κi and gi . Experiment with other set of parameters to observe how the prevalence within these patches will change. (c) Consider the same set of parameter values as given in (a) except that the parameters κ1 and N01 for Patch 1 can vary. Numerically plot the solutions for different values of these parameters and describe your observations. (d) We can also explore the effect of travel rates gi on the prevalence. Let β = 0.025 and fix all parameters as in part (b) except g1 and κ1 . Determine a couple of sets of parameter values of g1 and κ1 that can determine whether or not the infection on Patch 1 can go extinct.

474

14 Spatial Structure in Disease Transmission Models

Fig. 14.3 Time plots of the susceptibles (solid) and infectives (dashed) individuals for the resident populations 1 (thicker curve), 2 (intermediate), and 3 (thinner curve). The parameter values used are β = 0.025, κ1 = 3, κ2 = 2, κ3 = 1, g1 = 0.01, g2 = 0.02, g3 = 0.03 in (A) and g1 = 0.07, g2 = 0.03, g3 = 0.04 in (B). All other parameters have the same values as in part (A)

(A)

(B)

14.4 Project: A Patch Model with Residence Time Consider a model consisting of (14.7) and (14.8) with parameters N1 = N2 = 1,000,000,

β1 = 0.3056,

β2 = 0.1, γ = 1/6.5.

Question 1 Begin with an assumption that mixing is symmetric, p12 = p21 . Calculate the final epidemic size with several choices of p11 = p22 . Question 2 Calculate the effect on epidemic size of assuming no travel to the patch with a higher contact rate by assuming p22 = 1, p21 = 0 for several choices of p12 . Question 3 Calculate the effect on epidemic size of banning all travel, p12 = p21 = 0.

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Chapter 15

Epidemiological Models Incorporating Mobility, Behavior, and Time Scales

15.1 Introduction The work of Eubank et al. [24], Sara del Valle et al. [44], Chowell et al. [7, 18], and Castillo-Chavez and Song [13] have highlighted the impact of modified modeling approaches that incorporate heterogeneous modes of mobility within variable environments in order to study their impact on the dynamics of infectious diseases. Castillo-Chavez and Song [13], for example, proceeded to highlight a Lagrangian perspective, that is, the use of models that keep track at all times of the identity of each individual. This approach was used to study the consequences of deliberate efforts to transmit smallpox in a highly populated city, involving transient subpopulations and the availability of massive modes of public transportation. Here, a multi-group epidemic Lagrangian framework where mobility and the risk of infection are functions of patch residence time and local environmental risk is introduced. This Lagrangian approach has been used within classical contact epidemiological (that is, transmission is due to “contacts” between individuals) formulations in the context of a possible deliberate release of biological agents [2, 13]. The Lagrangian approach is introduced here as a modeling approach that explicitly avoids the assignment of heterogeneous contact rates to individuals. The use of contacts or activity levels and the view that transmission is due to collisions between individuals has a long history and it is conceptually consistent with the way we envision disease transmission between susceptible and infectious individuals. However, contacts are hard to define and consequently, at least in the context of communicable diseases, impossible to measure in various settings. Is it possible to capture interactions of individual mathematically in a way different from the notion of contacts? The approach that is proposed focuses on the use of modeling frameworks that involve patches/environments defined or characterized by risks of infection that are functions of the time spent in each environment/patch. These patches/environments may or may not have permanent hosts and they may be used to account for places of “transitory” residence like mass transportation systems or © Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_15

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hospitals or forests, to name but a few possibilities. Each environment or patch is characterized by the expected risk of infection of visitors as a function of time spent in such an environment. For example, a population near a forest may have some of its individuals spend time in the forest. Those who like the outdoors may be exposed longer to vectors than those who do not visit the forest. Consequently, the possibility of acquiring a vector-borne disease is a function of, among other factors, how long an individual spends in the forest each day. Similarly, individuals that use mass transportation routinely (during rush hour) are at a higher risk of acquiring a communicable disease including common colds, and it makes sense to assume that the risk may be a function of how long each individual spends each day commuting to work or to school. In other words, the average time spent in a community defined as a collection of environments that determined a priori the risk of acquiring an infection is at the heart of the Lagrangian approach. What is the Lagrangian approach and what does the theory tell us about the dynamics of such models in epidemic settings? We revisit this framework in possibly the simplest general setting that of a susceptible–infected–susceptible (SIS) epidemic multi-group model. We collect some of the mathematical formulae and results in the context of this general SIS multi-group model as reported in the literature [4, 6, 7, 11]. We proceed to identify basic reproduction numbers R0 as a function of the associated multi-patch residence-time matrix P (pi,j : i, j = 1, 2, 3 . . . n), which determines the proportion of time that a resident of Patch i spends in environment j . The analysis shows that the n-patch SIS model (as long as it is a strongly connected system) has a unique globally stable endemic equilibrium when R0 > 1, and a globally stable disease-free equilibrium when R0 ≤ 1. We have used simulations to generate insights on the impact that the residence matrix P has on infection levels within each patch. Model results [4, 6, 7, 11] show that the infection risk vector, which characterizes environments by risk to a pre-specified disease (measured by B), and the residence-time matrix P both play an important role in determining, for example, whether or not endemicity is reached at the patch level. Further, it is shown that the right combinations of environmental risks (B) and mobility behavior (P) are capable of promoting or suppressing infection within particular patches. The theoretical results [4, 6, 7, 11] are used to characterize patchspecific disease dynamics as a function of the time spent by residents and visitors in patches of interest. These results have helped classify patches as sources or sinks of infection, depending, of course, on the risk (B) and mobility (P) matrices. In general a residence-time matrix P cannot be made of constant entries in realistic settings. In fact the entries of P may depend on disease prevalence levels. We have explored some simple situations, via simulations, when the entries of the matrix P are state-dependent [4, 6, 7, 11]. The analysis and simulations for specific diseases are illustrated later in this chapter. They are used to highlight some of the possible differences that arise from having a state-dependent residence-time matrix P.

15.2 General Lagrangian Epidemic Model in an SIS Setting

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15.2 General Lagrangian Epidemic Model in an SIS Setting The following SIS model involving n-patches (environments) is introduced in [7]: Si′ = bi − di Si + γi Ii − Si λi (t)

(15.1)

Ii′ = Si λi (t) − γi Ii − di Ii

Ni′ = bi − di Ni ,

where bi , di , and γi denote the per-capita birth, natural death, and recovery rates, respectively, for i = 1, 2, 3 . . . n. The infection rates λi (t) have the form: λi (t) =

n j =1

n pkj Ik βj pij nk=1 , k=1 pkj Nk

i = 1, 2, . . . , n,

(15.2)

where pij denotes the proportion of susceptibles from Patch i who are currently in Patch j , βj is the risk of infection in Patch j , and the last fraction represents the proportion of infected in Patch j . Using the approach of the next generation matrix, the basic reproduction number R0 can be derived using the following system: I˙i =



 bi − Ii λi (t) − (γi + di )Ii , di

i = 1, 2, . . . , n.

As shown in the next section, R0 is a function of the risk vector B = (β1 , β2 , . . . , βn )t and the residence times matrix P = (pij ), i, j = 1, . . . , n, and it is shown in [7] that whenever P is irreducible (patches are strongly connected), the disease-free steady state is globally asymptotically stable if R0 ≤ 1 and a unique interior equilibrium exists and is globally asymptotically stable if R0 > 1. While a specific formula for the multi-patch basic reproduction number cannot be computed explicitly, it is possible in this case to find expressions for the patchspecific basic reproduction number. In fact, we have R0i (P) = R0i ×

n

pj i ,

j =1

where R0i are the local basic reproduction numbers (i = 1, 2, 3, · · · , n) computed when the patches are isolated from each other. From the R0i (i = 1, 2, 3, · · · , n), β the role that the relative risk that each environment (patch) plays, namely βji , can be assessed. Further, the role that residence times play in keeping track of the appropriate fraction of the population involved in a given patch is given by p b /d

(n ij i i ) . In other words, this patch specific R0i (i = 1, 2, 3, · · · ,) captures k=1 pkj bk /dk the impact of the P and B matrices.

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In short, if R0i (P) > 1 the disease persists in Patch i and furthermore, if pkj = 0 for all k = 1, 2, · · · , n and k = i whenever pij >1 , then it is shown that the disease dies in Patch i if R0i (P) < 1, that is, patch-specific basic reproduction numbers help characterize disease dynamics at the patch level [7]. We can look first at the following example of a multi-patch SIR model for a single outbreak: Si′ = −Si λi (t), Ii′ = Si λi (t) − αi Ii , Ri′ = αi Ii , i = 1, 2,

(15.3)

where Si , Ii , and Ri denote the population of susceptible, infected, and recovered immune individuals, respectively, in Patch i, and Ni = Si + Ii + Ri . This model is the same as the model (14.7–14.8) in the preceding chapter. The parameter αi denotes the per-capita recovery rate in Patch i and λi (t) are given in (15.2). In the rest of this chapter we make use of this Lagrangian modeling perspective (disease-specific versions) to carry out preliminary studies, in rather simple set ups, of the role of mobility in reducing or enhancing the transmission of specific diseases in regions of variable risk for the case of two patches. Numerical results are used to illustrate the power and limitations of this approach. Lagrangian models are used to explore the role that mobility plays in disease transmission for the cases of Ebola, tuberculosis, and Zika in simplified settings. Figure 15.1 represents a schematic representation of the Lagrangian dispersal between two patches.

Fig. 15.1 Dispersal of individuals via a Lagrangian approach

15.3 Assessing the Efficiency of Cordon Sanitaire as a Control Strategy of Ebola

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15.3 Assessing the Efficiency of Cordon Sanitaire as a Control Strategy of Ebola During the 2014 Ebola Epidemic in West Africa, it was observed [10, 46] that the rate of growth of the Ebola epidemic seemed to be increasing rather than decreasing as is standard in the study of epidemics. In other words, the reproduction number tends to decrease in time rather than increase. The evidence provided by the data and our analysis indicated that something was not right. We learned that troops were being used to prevent individuals from moving out of the most devastated communities facing Ebola. The use of cordons sanitaires seemed to be implemented even though past experiences have shown them to have a deleterious effect. Here, we formulate a two-patch mathematical model for Ebola virus disease (EVD) dynamics to highlight the potential lack of effectiveness or the deleterious impact of impeding mobility (cordons sanitaires). Via simulations, we look at the role of mandatory mobility restrictions and their impact on disease dynamics and epidemic final size. It is shown that mobility restrictions between high and low risk areas of closely linked communities are likely to have a deleterious impact on overall levels of infection in the total population involved.

15.3.1 Formulation of the Model The community of interest is assumed to be composed of two adjacent regions facing highly distinct levels of EVD infection and having access to a highly differentiated public health system (the haves and have-nots). There are differences in population density, availability of medical services, and isolation facilities. The need of those in the high-risk area to travel to the low-risk area is high as the jobs are in the well-off community. For Ebola, it may be unrealistic to assume susceptibles and infectives travel at the same rate. We let N1 denote the population in Patch 1 (high risk) and N2 the population in Patch 2 (low risk). The classes Si , Ei , Ii , Ri represent the susceptible, exposed, infective, and recovered sub-populations in Patch i (i = 1, 2). The class Di represents the number of disease induced deaths in Patch i. The dispersal of individuals is modeled via the Lagrangian approach defined in terms of residence times [4, 7]. The numbers of new infections per unit of time are based on the following assumptions: • The density of infected individuals mingling in Patch 1 at time t, who are only capable of infecting susceptible individuals currently in Patch 1 at time t, that is, the effective infectious proportion in Patch 1 is given by p11 I1 + p21 I2 , p11 N1 + p21 N2

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where p11 denotes the proportion of time that residents from Patch 1 spend in Patch 1 and p21 the proportion of time that residents from Patch 2 spend in Patch 1. • The number of newly infected Patch 1 residents while sojourning in Patch 1 is therefore given by β1 p11 S1



p11 I1 + p21 I2 p11 N1 + p21 N2



.

• The number of new infections within members of Patch 1, in Patch 2 per unit of time is therefore   p12 I1 + p22 I2 β2 p12 S1 , p12 N1 + p22 N2 where p12 denotes the proportion of time that residents from Patch 1 spend in Patch 2 and p22 the proportion of time that residents from Patch 2 spend in Patch 2. Hence, the effective density of infected individuals in Patch j is given by p1j N1 + p2j N2 ,

j = 1, 2.

If we further assume that infection by dead bodies occurs only at the local level (bodies are not moved), then, by following the same rationale as in model (15.3), we arrive at the following model: Di , Ni Di − κEi , Ei′ = Si λi (t) + εi βi pii Si Ni Ii′ = κEi − γ Ii , Di′ = fd γ Ii − νDi , Ri′ = (1 − fd )γ Ii + νDi , Ni = Si + Ei + Ii + Di + Ri , i = 1, 2, Si′ = −Si λi (t) − εi βi pii Si

(15.4)

where λi (t) are given in (15.2).

15.3.2 Simulations Simulations show that if only individuals from the high-risk region (Patch 1) were allowed to travel, then the epidemic final size can go under the cordon sanitaire level. Figure 15.2 captures the patch-specific prevalence levels for mobility values of p12 = 0, 0.2, 0.4, 0.6 with p21 = 0 (no movement). Disease dispersal, if the

15.3 Assessing the Efficiency of Cordon Sanitaire as a Control Strategy of Ebola

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Fig. 15.2 Dynamics of prevalence in each patch for values of mobility p12 = 0%, 20%, 40%, 60% and p21 = 0, with parameters: ε1,2 = 1.1, R01 = 2.45, R02 = 0.9, fd = 0.7, k = 1/7, ν = 1/2, γ = 1/7

Fig. 15.3 Dynamics of patch specific and total final epidemic size, for mobility values p12 = 0%, 20%, 40%, 60% and p21 = 0, with parameters: ε1,2 = 1.1, R01 = 2.45, R02 = 0.9, fd = 0.7, k = 1/7, ν = 1/2, γ = 1/7

disease spreads to a totally susceptible region, means that the secondary infections produced in the low-risk region reduce the overall two-patch prevalence, due to its access to better sanitary conditions and resources. However, there is a cost to the low-risk patch not only for the services provided but also for the generation of a larger number of secondary cases than if the “borders” were closed. Figure 15.3 shows that different mobility regimes can increase or decrease the total epidemic final size. In the presence of “low mobility” levels (p12 = 0.2, 0.4), the total final size curve may turn out to be greater than the cordon sanitaire case. We observe that the nonlinear impact of mobility on the total epidemic final size can bring it below the cordoned case even under relative “high mobility” regimes. This result highlights the trade-off that comes from reducing individuals’ time spent in a highrisk region versus exposing a totally susceptible population living in a safer region. Under certain mobility conditions, the results of such a trade-off are beneficial for the Global Commons.

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Fig. 15.4 Dynamics of maximum final size and maximum prevalence in Patch 1 with parameters: ε1,2 = 1.1, R01 = 2.45, R02 = 0.9, fd = 0.7, k = 1/7, ν = 1/2, γ = 1/7

Fig. 15.5 Dynamics of final epidemic size in the one way case with parameters: ε1,2 = 1.1, R01 = 2.45, R02 = 0.9, 1.0, 1.1, fd = 0.7, k = 1/7, ν = 1/2, γ = 1/7

Further, in order to clarify the effects of residence times on total final epidemic size, we proceeded to analyze its behavior under one way mobility. Figure 15.4 shows the cordon sanitaire (dashed gray line), patch specific, and the total epidemic final size for various possible mobility scenarios, p12 ∈ [0, 1]. We see that one way mobility reduces Patch 1 epidemic final size while increasing the Patch 2 final number of infections. We see that the total epidemic final size under low mobility (p12 < 0.5) is above the cordoned case. We also observe that Patch 2 sanitary conditions play an important role under high mobility regime bringing the total epidemic final size below the cordon sanitaire scenario (p12 > 0.5). Moreover, results suggest that for R02 < 1 extremely high mobility levels might eradicate an Ebola outbreak. It is important to stress that mobility reducing the total epidemic final size is dependent not only on the residence times and mobility type, but also on the patch-specific prevailing infection rates. Figure 15.5 shows that if R02 > 1 mobility is not capable of leading the total epidemic final size towards zero. Figure 15.6 shows that the global basic reproduction number decreases monotonically as one way mobility increases. However, it is not capable of capturing the harmful effect of low mobility levels, increasing the total epidemic final size. Mobility on its own is not always enough to reduce R0 below the critical

15.4 *Mobility and Health Disparities on the Transmission Dynamics of. . .

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Fig. 15.6 Dynamics of R0 with parameters: ε1,2 = 1, R01 = 2.45, R02 = 0.9, 1.0, 1.1, fd = 0.708, k = 1/7, α = 0, ν = 1/2, γ = 1/6.5

threshold. Instead, bringing the global R0 less than one requires reducing local risk, that is, getting a lower R02 .

15.4 *Mobility and Health Disparities on the Transmission Dynamics of Tuberculosis TB dynamics is the result of complex epidemiological and socio-economical interactions between and among individuals living in highly heterogeneous regional conditions. Many factors impact TB transmission and progression. A model is introduced to enhance the understanding of TB dynamics in the presence of diametrically distinct rates of infection and mobility. The dynamics are studied in a simplified world consisting of two patches, that is, two risk-defined environments, where the impact of short-term mobility and variations in reinfection and infection rates are assessed. The modeling framework captures “daily dynamics” of individuals within and between places of residency, work, or business. Activities are modeled by the proportion of time spent in environments (patches) having different TB infection risk. Mobility affects the effective population size of each Patch i (home of iresidents) at time t and they must also account for visitors and residents of Patch i, at time t. The impact that effective population size and the distribution of individuals’ residence times in different patches have on TB transmission and control is explored using selected scenarios where risk is defined by the estimated or perceived first time infection and/or exogenous reinfection rates. Model simulation results suggest that, under certain conditions, allowing infected individuals to move from high to low TB prevalence areas (for example, via the sharing of treatment and isolation facilities) may lead to a reduction in the total TB prevalence in an overall, twopatch, population.

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15.4.1 *A Two-Patch TB Model with Heterogeneity in Population Through Residence Times in the Patches Using a similar approach to model formulation, we consider the following model for the dynamics of TB in two patches: S˙i = μi Ni − Si λi (t) − μi Si , L˙ i = qSi λi (t) − Li λˆ i (t) − (γi + μi )Li + ρi Ii , I˙i = (1 − q)Si λi (t) + Li λˆ i (t) + γ Li − (μi + ρi )Ii ,

(15.5) i = 1, 2,

where λi (t) is the same as in (15.2) and λˆ i (t) =

2 pkj Ik , δj pij 2k=1 p k=1 kj Nk j =1

2

i = 1, 2.

15.4.2 *Results: The Role of Risk and Mobility on TB Prevalence We highlighted the dynamics of tuberculosis within a two-patch system, described by (15.5), under various residence times schemes via numerical experiments. The simulations were carried out using the two-patch Lagrangian modeling framework on pre-constructed scenarios. We assume that one of the two regions (say, Patch 1) has high TB prevalence. We do not model specific cities or regions. Nomenclature of some terms and scenarios are defined in Table 15.1. The interconnection of the two idealized patches demands that individuals from Patch 1 travel to the “safer” Patch 2 to work, to school, or for other social activities. It is assumed that the proportion of time that Patch 2 residents spend in Patch 1 is negligible. Here we define “high risk” based on the value of the probability of developing active TB using two distinct definitions: (i) patch having high direct first time transmission potential but no difference in exogenous reinfection potential between patches (β1 > β2 and δ1 = δ2 ) and (ii) the patch with high exogenous reinfection potential (δ1 > δ2 and β1 = β2 ). In addition, we assume a fixed population size for Patch 1 and vary the population size of Patch 2. Particularly, we assume that Patch 1 is the denser patch, while Patch 2 is assumed to have 12 N1 and 14 N1 . That is, contact rates are higher in the Patch 1 population as compared to corresponding rates in Patch 2.

15.4 *Mobility and Health Disparities on the Transmission Dynamics of. . .

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Table 15.1 Definitions and scenarios Nomenclature Risk High-risk patch

Enhanced socio-economic conditions (reducing health disparity) Mobility

Interpreted based on levels of infection rate, prevalence, or average contacts (via population size) Defined either by high direct first time infection rate (i.e., high β which leads to high corresponding R0 ) or by high exogenous reinfection rate (i.e., high δ) Defined by better healthcare infrastructure which is incorporated by high prevalence of a disease (i.e., high I (0)/N ) in a large population (i.e., large N )

Captured by average residence times of an individual in different patches (i.e., by using P matrix) Scenarios (assume high-risk and diminished socio-economic conditions in Patch 1 as compared to Patch 2) I1 (0) I2 (0) Scenario 1 β1 > β2 , δ 1 = δ 2 ; > , N1 > N2 ; N1 N2 vary p12 & p21 ≈ 0 I2 (0) I1 (0) > , N1 > N2 ; Scenario 2 β1 = β2 , δ 1 > δ 2 ; N1 N2 vary p12 & p21 ≈ 0

Fig. 15.7 Effect of mobility in the case of different transmission rates 0.13 = β1 > β2 = 0.07 (which gives R01 = 1.5, R02 = 0.8) and δ1 = δ2 = 0.0026, on the endemic prevalence. The cumulative prevalence and prevalence for each patch using the following population size proportions N2 = 21 N1 (left figure) and N2 = 41 N1 (right figure) are shown here. The green horizontal dotted line represents the decoupled case (i.e., the case when there is no movement between patches)

15.4.3 The Role of Risk as Defined by Direct First Time Transmission Rates In this subsection, we explore the impact of differences in transmission rates between patches. Patch 1 is high risk (R01 > 1; obtained by assuming β1 > β2 ), while Patch 2 in the absence of visitors would be unable to sustain an epidemic (R02 < 1). In addition the effect of different population ratios N1 /N2 is explored.

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Figure 15.7 uses mobility values p12 as it looks at their impact on increases in cumulative two-patch prevalence. At the individual patch level, increase in mobility values reduces the prevalence in Patch 1 but increases the prevalence in Patch 2 initially and then decreases past a threshold value of p12 (see red and black curves in Fig. 15.7). That is, completely cordoning off infected regions may not be a good idea to control disease. However, the movement rate of individuals between highrisk infection region and low-risk region must be maintained above a critical value to control an outbreak. Thus, it is possible that when Patch 1 (riskier patch) has a bigger population size, then mobility may turn out to be beneficial; the higher the ratio in population sizes, the higher the range of beneficial “traveling” times.

15.4.4 The Impact of Risk as Defined by Exogenous Reinfection Rates Here, we focus our attention on the impact of exogenous reinfection on TB’s transmission dynamics when transmission rates are the same in both patches, β1 = β2 . In this scenario, we assume the disease in both patches have reached an endemic state, that is, R01 > 1 and R02 > 1. However, Patch 1 remains the riskier, due to the assumption that exogenous reactivation of TB in Patch 1 is higher than in Patch 2, δ1 > δ2 . Figure 15.8 shows the combined role of exogenous reinfection and mobility values when the population of Patch 1 is twice or four times the population of Patch 2. It is possible to see a small reduction in the overall prevalence, given for all mobility values from Patch 1 to Patch 2. Within this framework, parameters, and scenarios, our model suggests that direct first time transmission plays a central role

Fig. 15.8 Effect of mobility when risk is defined by the exogenous reinfection rates 0.0053 = δ1 > δ2 = 0.0026 and β1 = β2 = 0.1 (which gives R01 = R02 = 1.155), on the endemic prevalence. The cumulative prevalence and prevalence for each patch using the following population size proportions N2 = 21 N1 (left figure) and N2 = 14 N1 (right figure) are shown here. The green dotted line represents the decoupled case (i.e., the case when there is no movement between patches)

15.4 *Mobility and Health Disparities on the Transmission Dynamics of. . .

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Fig. 15.9 Effect of mobility and population size proportions on the global basic reproduction number R0 when 0.13 = β1 > β2 = 0.07 and δ1 = δ2 = 0.0026

in TB dynamics when mobility is considered. Although mobility also reduces the overall prevalence when exogenous reinfection differs between patches, its impact is small compared to direct first time transmission results. Finally, Fig. 15.9 shows the relationship between population densities and mobility (p12 ) with respect to the basic reproduction number R0 . In this case we only explore the first case: direct first time transmission heterogeneity and found out that in this case mobility could indeed eliminate a TB outbreak. According to the World Health Organization (WHO) [48], in 2014, 80% of the reported TB cases occurred in 22 countries, all developing countries. Efforts to control TB have been successful in many regions of the globe and yet we still see 1.5 million people die each year. And so, TB, faithful to its history [19], still poses one of the greatest challenges to global health. Recent reports suggest that established control measures for TB have not been adequately implemented, particularly in subSaharan countries [1, 15]. In Brazil rates have decreased with relapse being more important than reinfection [20, 33]. Finally, in Cape Town, South Africa, a study [47] showed that in high incidence areas, individuals who have received TB treatment and are no longer infectious are at the highest risk of developing TB instead of being the most protected. Hence, policies that do not account for population specific factors are unlikely to be effective. Without a complete description of the attributes of the community in question, it is almost impossible to implement successful intervention programs capable of generating low reinfection rates through multiple pathways and low number of drug resistant cases. Intervention must account for the risks that are inherent with high levels of migration as well as with local and regional mobility patterns between areas defined by high differences in TB risk. This discussion of TB dynamics within a simplified framework of a two-patch system has captured in a rather dramatic way the dynamics in two worlds: the world of the haves and the world of the have-nots. Simulations of simplified extreme scenarios highlight the impact of disparities. TB dynamics depend on the basic reproduction number (R0 ), a function of model parameters that includes direct first transmission and exogenous (reinfection)

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transmission rates. The simulations of specific extreme scenarios suggest that shortterm mobility between heterogeneous patches does not always contribute to overall increases in TB prevalence. The results show that when risk is considered only in terms of exogenous reinfection, global TB prevalence remains almost unchanged when compared to the effect of direct new infection transmission. In the case of a high-risk direct first time transmission, it is observed that mobile populations may contribute to prevalence levels in both environments (patches). The simulations show that when the individuals from the risky population spend 25% of their time or less in the safer patch this is bad for the overall prevalence. However, if they spend more, the overall prevalence decreases. Further, in the absence of exogenous reinfections, the model is robust, that is, the disease dies out or persists based on whether or not the basic R0 is below or above unity, respectively. Although, the role of exogenous reinfection seems not that relevant to overall prevalence, the fact remains that such mode of transmission increases the risk that comes from a large displacement of individuals into a particular TB-free areas, due to catastrophes or conflict. As noted in [25], ignoring exogenous reinfections, that is, establishing policies that focus exclusively on the reproduction number R0 , would amount to ignoring the role of dramatic changes in initial conditions, now more common than before, due to the displacement of large groups of individuals, the result of catastrophes, and/or conflict.

15.5 *ZIKA In November 2015, El Salvador reported their first case of Zika virus (ZIKV), an event followed by an explosive outbreak that generated over 6000 suspected cases in a period of 2 months. National agencies began implementing control measures that included vector control and recommending an increased use of repellents. In addition in response to the alarming and growing number of microcephaly cases in Brazil, the importance of avoiding pregnancies for 2 years was stressed. The role of mobility within communities characterized by extreme poverty, crime, and violence where public health services are not functioning is the set up for this example. We use a Lagrangian modeling approach within a two-patch setting in order to highlight the possible effects that short-term mobility, within two highly distinct environments, may have on the dynamics of ZIKV when the overall goal is to reduce the number of cases in both patches. The results of simulations in highly polarized and simplified scenarios are used to highlight the role of mobility on ZIKV dynamics. We found that matching observed patterns of ZIKV outbreaks was not possible without incorporating increasing levels of heterogeneity (more patches). A lack of attention to the threats posed by the weakest links in the global spread of diseases poses a serious threat to global health policies (see [12, 16, 23, 34, 40– 42, 50]). Our results highlight the importance of focusing on key nodes of global transmission networks, which in the case of many regions correspond to places where the level of violence is highest. Latin America and the Caribbean, which

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house 9% of the global population are a particular hot spot because this region accounts for 33% of the world’s homicides [29]. Hence, it is essential to assess how much public safety conditions may affect mobility and the level of local risk, which may affect the dynamics of ZIKV.

15.5.1 *Single Patch Model Assume that individuals while in Patch 1 will be experiencing high risk of infection, while those in Patch 2 will be experiencing low risk. Movement (daily activities) will alter the amount of time that each individual spends on each patch, the longer that an individual is found in Patch 1, the more likely that he/she will become infected. The level of patch-specific risk to infection is captured via the use of a single parameter βˆi , i = 1, 2 with βˆ1 ≫ βˆ2 . This assumption pretends to capture health disparities in a rather simplistic way. The case of Johannesburg and Soweto in South Africa, or North and South Bogota in Colombia, or Rio de Janeiro and adjacent favelas in Brazil, or gang-controlled and gang-free areas within San Salvador are but some of the unfortunately large number of pockets dominated by conflict, high crime or highly differentiated health structures within urban centers around world. The short time scale dynamics of individuals (going to work or attending schools or universities) are incorporated within this model. The dynamics of transmission is carried out via simulations over the duration of a single outbreak. The ZIKV dynamics single patch model involves host and vector populations of size Nh and Nv , respectively. Both populations are subdivided by epidemiological states; the transmission process is modeled as the result of the interactions of these populations. On that account, we let Sh , Eh , Ih,a , Ih,s , and Rh denote the susceptible, latent, infectious asymptomatic, infectious symptomatic, and recovered host sub-populations. Similarly, Sv , Ev , and Iv are used to denote the susceptible, latent, and infectious mosquito sub-populations. Since the focus is on the study of disease dynamics over a single outbreak, we neglect the host demographics while assuming that the vector demographics do not change, meaning that it is assumed the birth and death per-capita mosquito rates cancel each other out. New reports [14, 21] point out the presence of large numbers of asymptomatic ZIKV infectious individuals. Consequently, we consider two classes of infectious Ih,a and Ih,s , that is, asymptomatic and symptomatic infectious individuals. Further, since there is no full knowledge of the dynamics of ZIKV transmission, it is assumed that Ih,a and Ih,s individuals are equally infectious with their periods of infectiousness roughly the same. Our assumptions could be used to reduce the model to one that considers a single infectious class Ih = Ih,a + Ih,s . We keep both infectious classes as it may be desirable to keep track of each type. These assumptions may not be too bad given our current knowledge of ZIKV epidemiology and the fact that ZIKV infections, in general, are not severe. Furthermore, given that the infectious process of ZIKV is somewhat similar to that of dengue, we use some of the parameters estimated in dengue transmission studies within El Salvador. ZIKV basic reproduction number

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estimates are taken from those that we just estimated using outbreak data from Barranquilla Colombia [45]. Furthermore, the selection of model parameters ranges used also benefited from prior estimates conducted with data from the 2013–2014 French Polynesia outbreak [31], some of the best available. The dynamics of the prototypic single patch system, single epidemic outbreak, can therefore be modeled using the following standard nonlinear system of differential equations [9]: Sh′ = −bβvh Sh NIvh Eh′ = bβvh Sh NIvh − νh Eh ′ = (1 − q)ν E − γ I Ih,s h h h h,s ′ = qν E − γ I Ih,a h h h h,a Rh′ = γh (Ih,s + Ih,a ) I +I Sv′ = μv Nv − bβhv Sv h,sNh h,a − μv Sv I

(15.6)

+I

Ev′ = bβhv Sv h,sNh h,a − (μv + νv )Ev Iv′ = νv Ev − μv Iv .

15.5.2 *Residence Times and Two-Patch Models The role of mobility between two communities, within the same city, living under dramatically distinct health, economic, social, and security settings is explored using a model as simple as possible, that is, a model that only considers two patches (prior modeling efforts that didn’t account for the effective population size but that incorporated specific controls include [32]). Patch 2 has access to working health facilities, low crime rate, adequate human and financial resources, and adequate public health policies, in place. Patch 1 lacks nearly everything and crime is high. The differences in risk are captured by postulating very different transmission rates. We study the dynamics of host mobility in highly distinct environments, with risk ˆ Hence, βˆ1 ≫ βˆ2 , where βˆi defines the being captured by the transmission rate, β. risk in Patch i, i = 1, 2 [Patch 1 (high risk) and Patch 2 (low risk)]. The host populations are stratified by epidemiological classes indexed by the patch of residence. In particular, Sh,i , Eh,i , Ih,a,i , Ih,s,i , and Rh,i denote the susceptible, latent, infectious asymptomatic, infectious symptomatic, and recovered host populations in Patch i, i = 1, 2 with Sv,i , Ev,i , and Iv,i denoting the susceptible, latent, and infectious mosquito populations in Patch i, i = 1, 2. As before, Nh,i denotes the host patch population size (i, i = 1, 2) and Nv,i the total vector population in Patch i, i = 1, 2. The vector is assumed to be incapable of moving between patches, a reasonable assumption in the case of Aedes aegypti under the appropriate spatial scale. The patch model parameters are presented in Table 15.2 with the flow diagram, single patch dynamics model, capturing the situation when residents and visitors do not move; that is, when the 2 × 2 residence times matrix P is such that p11 = p22 = 1 (Fig 15.10).

15.5 *ZIKA

493

Table 15.2 Description of the parameters used in system (15.6) Parameters βvh βhv bi

Description Infectiousness of human to mosquitoes Infectiousness of mosquitoes to humans Biting rate in Patch i

Value 0.41 0.5 0.8

νh

Humans’ incubation rate

1 7

q

Fraction of latent that become asymptomatic and infectious

0.1218

γi

Recovery rate in Patch i

1 5

pij

Proportion of time residents of Patch i spend in Patch j

[0, 1]

μv

Vectors’ natural mortality rate

1 13

νv

Vectors’ incubation rate

1 9.5

Fig. 15.10 Flow diagram of model (15.6)

Since individuals experience a higher risk of ZIKV infection while in Patch 1, then it is assumed that mobility from Patch 2 to Patch 1 is unappealing with typical Patch 2 residents spending (on the average) a reduced amount of time each unit of

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15 Epidemiological Models Incorporating Mobility, Behavior, and Time Scales

time in Patch 1. Parameters are chosen so that the dynamics of ZIKV within Patch 2 cannot be supported in the absence of mobility between Patch 1 and Patch 2. Thus, the Patch 2 local basic reproduction number is taken to be less than one, namely R02 = 0.9. Mobility is modeled under the residence times matrix P with entries given initially by p21 = 0.10 and p12 = 0. Two cases are explored: A “worst case” scenario where control measures are hardly implemented due to crime, conflict, or other factors on Patch 1, that is, Patch 1 is a place where the risk of acquiring a ZIKV infection is high since R01 = 2. The “best case” scenario corresponds to the case when Patch 1 can implement some control measures with some degree of effectiveness, and consequently Patch 1 has an R01 = 1.52. The R0i values used are in line with those previously estimated for ZIKV outbreaks [31, 45]. Simulations are seeded by introducing an asymptomatic infected individual in Patch 1 under the assumption that the host and vector populations are fully susceptible in both patches. Figure 15.11 (top) shows the incidence and final ZIKV epidemic size when Patch 1 is under the “worst case scenario,” defined by a basic reproduction number of

Infected Patch 1

Final Size Patch 1

0.06 0.96

p12 = 0% 0.05

p p

0.04

p

12 12 12

= 15%

0.959

= 30%

0.958

= 45%

0.957 0.956

0.03

0.955 0.02

p

0.954 0.953

= 0%

p12 = 30%

0.01 0.952 0

12

p12 = 15% p12 = 45%

0.951 0

100

200

300

400

Time

500

600

700

200

300

400

500

600

700

Time

Fig. 15.11 Per patch incidence and final size proportions for p21 = 0.10, p12 = 0, 0.15, 0.30, and 0.45. Mobility shifts the behavior of the Patch 1 final size in the “worst case” scenario: R01 = 2 and R02 = 0.9

15.5 *ZIKA

495

R01 = 2 [45]. Figure 15.11 shows that at p12 = 0.15 the final number of infected residents in Patch 1 is larger to the number in the baseline scenario (p12 = 0). In fact, it reaches almost 96% of the population, an unrealistic value. Additional simulated p12 values show that final Patch 1 size would go below the baseline case; a benefit of mobility. Figure 15.11 highlights the case when the Patch 2 epidemic final size grows with increases in mobility when compared with the baseline case (no mobility from Patch 1). We see reductions in the Patch 1 epidemic final size for some mobility values accompanied by increments in the Patch 2 epidemic final size when compared to the baseline scenario (no mobility from Patch 1). Specifically, reductions in Patch 1 epidemic final size are around 1 × 10−3 , while increments in Patch 2 are around 1 × 10−2 , under the assumption that the population in Patch 1 is the same as that in Patch 2. Thus while mobility may provide benefits within Patch 1 (under the above assumptions) the fact remains that it does it at a cost. In short, it is also observed that the epidemic final size per patch does not respond linearly to changes in mobility even when only the mobility p12 is increased (see Figs. 15.11 and 15.12).

Fig. 15.12 Per patch incidence and final size proportions for p21 = 0.10, p12 = 0, 0.15, 0.30, and 0.45. Mobility significantly shapes the per patch final sizes in the “worst case” scenario R01 = 2 and R02 = 0.9

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15 Epidemiological Models Incorporating Mobility, Behavior, and Time Scales

Consider now the “best case” scenario, a basic reproduction number R01 = 1.52, under the assumption that the population in Patch 1 is the same as that in Patch 2. The results of simulations collected in Fig. 15.12 show a final size epidemic curve similar to that generated in the “worst case” scenario for Patch 1. Some mobility values can increase the Patch 1 epidemic final size, reaching almost 80% of the population when p12 = 0.30, an unrealistic level, albeit, as expected lower than in the “worst” case scenario. The existence of a mobility threshold after which the final epidemic sizes in Patch 1 start to decrease is also observed. The results in Fig. 15.12 suggest that under all p12 mobility levels, Patch 2 ZIKV epidemic final size supports monotone growth in the total number of infected individuals. The changes in the epidemic final size in each patch in Fig. 15.12 are roughly equivalent (the same order, 1 × 10−2 ) given that the population in Patch 1 is the same as that in Patch 2. The simulation results presented so far provide only partial information on the impact that short-term mobility may have on the transmission dynamics of ZIKV. Now, by fixing the mobility from Patch 2 to Patch 1, we are just focusing only on the impact of changes in mobility from Patch 1 to Patch 2. Further, the potential changes in mobility patterns that host populations may have in response to ZIKV dynamics are ignored by our use of a mobility matrix P with constant entries pij . We found that epidemic final size within Patch 1 is qualitatively similar in the worst and best case scenarios: increasing at first, decreasing after a certain threshold, and crossing down the baseline case under some mobility regimes. Further, it has been observed that the qualitative behavior of the epidemic final size in Patch 2 grows monotonically as mobility increases. Patch 1 and Patch 2 responses are of different orders of magnitude in the “worst case” scenario but roughly of the same order of magnitude in the “best case” scenario, which means, under our restrictive conditions and assumptions, that reductions in risk in Patch 1 do help significantly.

15.5.2.1

*The Role of Risk Heterogeneity in the Dynamics of ZIKV Transmission

The impact of risk heterogeneity on ZIKV dynamics within the overall twopatch system is explored, an analysis that requires the numerical estimation of the global reproduction number as a function of the mobility matrix P. Using the previous scenarios (R01 = 1.52, 2) simulations are carried out first assuming equal population sizes (N1 = N2 ). However, when looking at the impact of changes in risk on Patch 2 (R02 = 0.8, 0.9, 1, 1.1), our simulations identify a growing epidemic in Patch 2 as risk increases with the overall community experiencing nonlinear changes in risk as residency times change from the baseline scenario given by p12 = 0. Specifically, Fig. 15.13 captures overall reductions on the global reproduction number for all residence times while identifying the existence of a residence time interval for which mobility decreases the total size of the outbreak in the two-patch community, when compared to the corresponding baseline case (p12 = 0). In the absence of mobility from Patch 1 (p12 = 0), increases in the epidemic final size as R0i increases are observed. These simulations show that

15.5 *ZIKA

497

Fig. 15.13 Local and global final sizes through mobility values when p21 = 0.10. Although mobility reduces the global R0 , allowing mobility in the case of El Salvador (R0 = 2) might lead to a detrimental effect in the global final size

mobility can slow down the speed of the outbreak (smaller global R0 ). Of course, the simulation results also re-affirm the obvious, that is, that the existence of a high risk, mobile, and well-connected patch can serve as an outbreak magnifier; a situation that has been explored within an n-patch system under various connective schemes [7, 10]. This is because, in the two-patch case, it is observed that the global reproduction number R0 experiences reductions for almost all mobility values. For the scenarios selected R0 never drops below 1. Hence, under our assumptions and scenarios, it is seen that the use of fixed mobility patterns makes the elimination of ZIKV extremely difficult if not impossible under our two scenarios. Figure 15.13 provides an example that highlights the relationship between the global reproduction number and corresponding epidemic final size.

15.5.2.2

*The Role of Population Size Heterogeneity in the Dynamics of ZIKV Transmission

The role of population density in the total epidemic final size and global basic reproduction number is explored under our two scenarios, now under the changed assumption that the densities (population sizes) of Patch 1 and Patch 2 differ. Specifically, we take N1 = 2N2 , 3N2 , 5N2 , and 10N2 .

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15 Epidemiological Models Incorporating Mobility, Behavior, and Time Scales

Fig. 15.14 Total final size and global basic reproduction number through mobility values when p21 = 0.10. Local risk values are set up to R02 = 0.9 and R01 = 1.52, 2

It is observed that difference in population sizes do matter. Specifically, it is observed that (under our selections) a big difference in density indicate that a higher epidemic final size is reached. The value of 90% for the “worst case” is possible with changes in the global reproduction number exhibiting different patterns (see Fig. 15.14). We observe that despite increases in the total epidemic final size as mobility changes the global R0 actually decreases monotonically for most residence times, never falling below one. A sensible degree of magnification on the spread of the disease as residence times change is observed whenever the differences between N1 and N2 are not too extreme. In fact, it is possible for mobility to be beneficial in the control of ZIKV under the above simplistic extreme scenarios. Simulations continue to show that under the prescribed conditions and assumptions, model generated ZIKV outbreaks remain unrealistically high. The simulations show, for example, that the global reproduction number reaches its minimum at around p12 = 0.90 with Fig. 15.14 showing that the larger the high risk population gets (N1 >> N2 ), the greater the total epidemic final size becomes as individuals from Patch 1 spend more than half of their day in Patch 2. Using a low p12 value, a small benefit is observed, namely the total epidemic final size is reduced, when the differences between R0i are high. For the two epidemiological scenarios R01 = 2 and R01 = 1.52, Tables 15.3 and 15.4 provide a summary of the average proportion of infected for low

15.5 *ZIKA

499

Table 15.3 Final size (Patch 1, Patch 2) N1 = 10,000, R01 = 2, R02 = 0.9, and p21 = 0.10 N2 N1 N1 N1 N1 N1

= N2 = 2N2 = 3N2 = 5N2 = 10N2

Low mobility (0.9594, 0.5333) (0.9683, 0.5418) (0.9709, 0.5390) (0.9729, 0.5283) (0.9741, 0.5030)

Intermediate mobility (0.9583, 0.5633) (0.9685, 0.5599) (0.9713, 0.5478) (0.9732, 0.5255) (0.9743, 0.4908)

High mobility (0.9539, 0.6122) (0.9667, 0.6116) (0.9701, 0.6018) (0.9725, 0.5852) (0.9739, 0.5624)

Min R0 1.4954 1.6786 1.7640 1.8457 1,9173

Table 15.4 Final size (Patch 1, Patch 2) N1 = 10,000, R01 = 1.52, R02 = 0.9, and p21 = 0.10 N2 N1 N1 N1 N1 N1

= N2 = 2N2 = 3N2 = 5N2 = 10N2

Low mobility (0.7920, 0.3756) (0.8287, 0.3938) (0.8398, 0.3948) (0.8480, 0.3877) (0.8533, 0.3652)

Intermediate mobility (0.7950, 0.4010) (0.8340, 0.4061) (0.8448, 0.3956) (0.8520, 0.3731) (0.8556, 0.3352)

High mobility (0.7849, 0.4304) (0.8300, 0.4356) (0.8422, 0.4248) (0.8500, 0.4046) (0.8542, 0.3756)

Min R0 1.1853 1.3023 1.3590 1.4141 1.4630

Fig. 15.15 Global R0 dynamics through mobility when p21 = 0.10. Patch 2 populations vary from N1 = N2 , 2N2 , 3N2 , 5N2 up to N1 = 10N2 . The global R0 hits its minimum always at an unrealistic 91% of mobility. As N1 approaches N2 , this minimum value decreases

(p12 = 0–0.2), intermediate (p12 = 0.2–0.4), and high mobility (p12 = 0.4–0.6) when p21 = 0.10. The role of population scaling N1 = 2N2 , 3N2 , 5N2 , and 10N2 is also explored. Figure 15.15 shows the global R0 over all mobility values for different population weights in the two epidemic scenarios. The minimum R0 value is reached for all cases when mobility is at an unrealistic 91% and when N1 ≈ N2 . The results collected in Fig. 15.15 show that short-term mobility plays an important role in ZIKV dynamics, again, under a system involving two highly differentiated patches. Simulations also suggest that, even though mobility can reduce the global reproduction number, mobility by itself is not enough to eliminate an outbreak or make a real difference under our two scenarios.

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15 Epidemiological Models Incorporating Mobility, Behavior, and Time Scales

15.5.3 What Did We Learn from These Single Outbreak Simulations? The study of the role of mobility at large spatial scales may be best captured using question-specific related models that account for the possibility of long-term mobility (see, for example, [2, 3, 18, 22, 27, 28, 30, 43]). Here, we made use of two patches, as distinct as they can be would be able to shed some light on the transmission dynamics of ZIKV, whenever extreme health disparities within neighboring communities or within urban centers were the norm. Although the goal is not to fit specific outbreaks, we decided to make use of recently published parameter ranges, including some reported by us [45]. The impact of ZIKV can be assessed locally (each patch) or globally, that is, over the two-patch system. Here, system risk assessment was carried out by computing R0 , via the numerical solution of a system of nonlinear equations. Changes in the system R0 were computed (as residence times were varied) in relationship to the local R0i , that is, local basic reproduction numbers (in the absence of mobility). Further, the mobility-dependent system epidemic final sizes were computed via simulations that assessed the impact of mobility (and risk) locally and on the overall system. The metrics used in our assessment included the overall epidemic final size (a measure of the overall impact of an outbreak), a function of mobility within the two selected scenarios (R01 = 1.52 and R01 = 2). The challenges posed by policies that may be beneficial to the system but detrimental to each patch were explored within our two-patch system. Situations where the total final epidemic size increased with increments in R02 , and situations where the total final epidemic size decreased under low mobility values for R(02) were documented. Population density does make a difference and examples when R02 < 1 with mobility incapable of reducing the total epidemic final size under no differences in patch density (here measured by total population size in each patch, both assumed to have roughly the same area) were also identified. Differences in population density were also shown to be capable of generating reductions on the total final epidemic size within some mobility regimes. The highly simplified two-patch model used seemed to have shed some light on the role of mobility on the spread of ZIKV in areas where huge differences in the availability of public health programs and services—the result of endemic crime, generalized violence, and neglect—exist. Model simulations seemed to have shed some light on the potential relevance of factors that we failed to account for. ˆ has strengths and The value of the use of single patch-specific risk parameters (β) limitations. The model used did not account explicitly for changes in the levels of infection within the vector population nor did it account for the impact of substantial differences in patch vector population sizes. The simplified model failed to account for the responses to outbreaks by patch residents as individuals may alter mobility patterns, use more protective clothing while responding individually and independently to official control programs in the face of dramatic increases on the vector population or due to a surge in the number of cases. The use of two patches

References

501

and severe assumptions limits the outcomes that such an oversimplified system can support. Communities can’t in generally be modeled under a highly differentiated two-tier system and in the case of ZIKV, the possibility of vertical transmission in humans and vectors as well as sexually transmitted ZIKV cannot be completely neglected [8, 39]. The introduction of changes in behavior in response to individuals’ assessment of the levels of risk infection over time needs to be addressed [10]; a challenge that has yet to be met to the satisfaction of the scientific community involved in the study of epidemiological processes as complex adaptive systems (see, for example, [26, 36, 42]). The limitations of the role of technology in the absence of the public health infrastructure—there is no silver bullet—have been addressed in the context of Ebola [16, 49]. It would be interesting to see the impact of technology in settings where health disparities are pervasive, using a two-patch Lagrangian epidemic model in the context of communicable and vector-borne diseases, including dengue, tuberculosis , and Ebola [5, 23, 34, 35]. Further, its often the case that the use of simplified models quite often overestimates the impact of an outbreak (see [37, 38]) and so find the right level of model heterogeneity (number of patches) becomes a pressing and challenging question. What is the right level of aggregation to address these questions? Certainly, we have seen the use of dramatic measures to limit the spread of diseases like SARS, influenza , or Ebola [16, 17, 27], as well as the rise of vectorborne diseases like dengue and Zika, and the dramatic implications that some measures have had on local and global economies. The question remains, what can we do to mitigate or limit the spread of disease, particularly emergent diseases without disrupting central components? Discussions on these issues are recurrent [26, 36], most intensely in the context of SARS, influenza, Ebola , and Zika, in the last decade or so. The vulnerability of world societies is directly linked to the lack of action in addressing the challenges faced by the weakest links in the system. This must be accepted and acted on by the world community. We need global investments in communities and nations where health disparities and lack of resources are the norm. We must invest in research and surveillance within clearly identified world hot spots, where the emergence of new diseases is most likely to occur. We must do so with involvement at all levels of the affected communities [12, 44].

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46. Towers, S., O. Patterson-Lomba, and C. Castillo-Chavez (2014) Temporal variations in the effective reproduction number of the 2014 West Africa Ebola outbreak. PLoS currents 6. 47. Verver, S., R.M. Warren, N. Beyers, M. Richardson, G.D. van der Spuy, M.W. Borgdorff, D. A. Enarson, M.A. Behr, and P.D. van Helden (2005) Rate of reinfection tuberculosis after successful treatment is higher than rate of new tuberculosis, Am. J. Respiratory and Critical Care Medicine 171: 1430–1435. 48. W. H. O. (WHO) (2015) Tuberculosis, fact sheet no. 104. Online, October 2015. 49. Yong, K., E.D. Herrera, and C. Castillo-Chavez (2016)From bee species aggregation to models of disease avoidance: The Ben-Hur effect. In Mathematical and Statistical Modeling for Emerging and Re-emerging Infectious Diseases, pages 169–185. Springer. 50. Zhao, H., Z. Feng, and C. Castillo-Chavez (2014) The dynamics of poverty and crime. J. Shanghai Normal University (Natural Sciences· Mathematics), pages 225–235.

Part IV

The Future

Chapter 16

Challenges, Opportunities and Theoretical Epidemiology

Lessons learned from the HIV pandemic, SARS in 2003, the 2009 H1N1 influenza pandemic, the 2014 Ebola outbreak in West Africa, and the ongoing Zika outbreaks in the Americas can be framed under a public health policy model that responds after the fact. Responses often come through reallocation of resources from one disease control effort to a new pressing need. The operating models of preparedness and response are ill-equipped to prevent or ameliorate disease emergence or reemergence at global scales [27]. Epidemiological challenges that are a threat to the economic stability of many regions of the world, particularly those depending on travel and trade [132], remain at the forefront of the Global Commons. Consequently, efforts to quantify the impact of mobility and trade on disease dynamics have dominated the interests of theoreticians for some time [14, 143]. Our experience includes an H1N1 influenza pandemic crisscrossing the world during 2009 and 2010, the 2014 Ebola outbreaks, limited to regions of West Africa lacking appropriate medical facilities, health infrastructure, and sufficient levels of preparedness and education, and the expanding Zika outbreaks, moving expeditiously across habitats suitable for Aedes aegypti. These provide opportunities to quantify the impact of disease emergence or reemergence on the decisions that individuals take in response to real or perceived disease risks [11, 62, 93]. The case of SARS in 2003 [40], the efforts to reduce the burden of H1N1 influenza cases in 2009 [33, 62, 80, 93] and the challenges faced in reducing the number of Ebola cases in 2014 [24, 27] are but three recent scenarios that required a timely global response. Studies addressing the impact of centralized sources of information [150], the impact of information along social connections [33, 37, 42], or the role of past disease outbreak experiences [105, 130] on the risk-aversion decisions that individuals undertake may help identify and quantify the role of human responses to disease dynamics while recognizing the importance of assessing the timing of disease emergence and reemergence. The co-evolving human responses to disease dynamics are prototypical of the feedbacks that define

© Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9_16

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complex adaptive systems. In short, we live in a socioepisphere being reshaped by ecoepidemiology in the “Era of Information”. What are the questions and modes of thinking that should be driving ongoing research on the dynamics, evolution, and control of epidemic diseases at the population level? The challenges of SARS, Ebola, Influenza, Zika, and other diseases are immense. While we may guess which emerging or re-emerging disease may lead to the next possible catastrophe, we cannot know. The contemporary philosopher Yogi Berra is rumored to have said, “Making predictions is hard, especially about the future”. There are some epidemiological topics that have already received some attention but are not yet fully developed. In the rest of this chapter we highlight some challenges, opportunities, and promising approaches in the study of disease dynamics at the population level.

16.1 Disease and the Global Commons As has been noted, “The identification of a theoretical explanatory framework that accounts for the pattern regularity exhibited by a large number of host–parasite systems, including those sustained by host–vector epidemiological dynamics, is but one of the challenges facing the co-evolving fields of computational, evolutionary, and theoretical epidemiology” [25]. Furthermore, “The emergence of new diseases, the persistence of recurring diseases and the re-emergence of old foes, the result of genetic changes or shifts in demographic, and environmental shifts have increased due to mobility, global connectedness, trade, bird migration, poverty and longlasting violent conflicts. These diseases often present modeling challenges which may yield to existing analytic techniques but sometimes require new mathematics” [25]. The Global Commons are continuously reshaped by the ability of an increasing proportion of the human population to live, move, or trade nearly anywhere. Therefore, understanding the patterns of interactions between humans, or between humans and vectors, as well as their relationships to patterns of individual movement, particularly those of the highly mobile, is critical to public health responses that effectively ameliorate the ability of a disease to spread. In today’s world, hosts’ risk knowledge (good or bad information) when combined with the ability of public health officials to measure and properly communicate, in a timely manner, real or perceived information on disease risks, limit our ability to derail the spread of emergent and re-emergent diseases, at time scales that make a difference.

16.1.1 Contagion and Tipping Points Contagion is believed to be the direct or indirect result of interactions between individuals experiencing radically different epidemiological, or immunological, or

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social states. Contagion tends to succeed within environments or communities that “facilitate” modes of infection among its members. Contagion is an “understood” or “believed” mode of disease transmission or of “socially transmitted” behaviors, popularized by Malcolm Gladwell, a journalist who made use of his general understanding of the concept of “contagion.” In his construction of reasonable or plausible explanations for the observed and documented dramatic reductions in car thefts and violent crimes in New York City in the 1990s, [71] Gladwell expanded the use of the concept of contagion and tipping point in his development of a framework that captures—as the result of contagion—the spread of a multitude of social ills or virtues [72]. Specifically, contagion is seen in [71] as a force capable of starting and sustaining growth in criminal activity as long as a “critical mass” of individuals capable and willing to commit crimes is available. The growth in criminal activity in New York City is, according to Gladwell, the result of the “interactions” between a large enough pool of criminally active (infected) individuals and individuals susceptible to criminal contagion [71]. Gladwell extends the perspective pioneered by Sir Ronald Ross [141] and his “students” [90–92] to the field of social dynamics. Gladwell concludes as Ross did in 1911 that implementing control measures (crime contact-reduction measures) that bring the size of the population of criminals (the core) below a critical threshold (tipping point), are sufficient to explain the drastic reductions in criminal activity in NYC. Gladwell concludes, “There is probably no other place [NYC] in the country where violent crime has declined so far, so fast” [71].

16.1.2 Geographic and Spatial Disease Spread The SARS epidemic of 2002–2003 emphasized the possibility of disease transmission over long distances through air travel, and this has led to metapopulation studies that account for long-distance transmission [5–8]. A metapopulation, in this context, is a population of populations linked by travel. A metapopulation model would have an associated, independent of travel, reproduction number as well as reproduction numbers that account for travel between patches, either temporary travel or permanent migration . This is an Eulerian perspective, describing migration between patches. An alternative approach to the modeling of the spatial spread of diseases is based on a Lagrangian perspective, which can be formulated, for example, in terms of residence times [18, 25] . This approach has been introduced in Chap. 15. In this structure, actual travel between patches is not described explicitly, and this makes the analysis less complicated. Calculation of the reproduction number and the final size relations is possible. Another aspect of the study of the spread of diseases is the spatial spread of diseases through diffusion. This has been introduced in Chap. 14 of this book and has been examined in considerable detail in [136], with particular emphasis on epidemic waves.

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16.2 Heterogeneity of Mixing, Cross-immunity, and Coinfection In epidemics, as in the rest of biology, the role of heterogeneity plays a fundamental role and a critical question arises: what is the level of heterogeneity that must be included to address a specific question properly? For example, first-order estimates of the fraction that must be vaccinated to eliminate a communicable disease can be handled with homogeneous mixing models while the elaboration of optimal vaccination strategies in real-life situations often require an age-structured model [28, 81]. The study of nosocomial (in-hospital) infections provides an additional example of the role of heterogeneity in transmission or degree of susceptibility or resistance [38, 39, 106]. The SARS epidemic provided a timely example of the criticality of heterogeneous mixing, in nosocomial transmission [85, 152] . Since there was no treatment available during the SARS epidemic, the main management approach rested on the effectiveness of isolation of diagnosed infectives, quarantine of suspected infectives, and early diagnosis. Quarantine was decided by tracing of contacts made by infectives but in fact few quarantined individuals developed SARS symptoms. The role of early diagnosis and the effectiveness of isolation seemed to have been the key to SARS control with improvements in contact tracing also playing an important role in epidemic control. Another set of questions arises when one considers the immunological history of individuals or populations. There are many instances in which more than one strain of a disease is circulating within a population and the possibility of cross-immunity between strains becomes important [3, 4]. Mathematically, co-strain co-circulation may lead to models that support a disease-free equilibrium (or non-uniform age distribution), equilibria in which only one strain persists, and an equilibrium in which two strains coexist. The role of cross-immunity in destabilizing disease dynamics (periodic solutions) has been studied extensively in the case of influenza models without age structure [57, 123, 124, 151] and also in age-structured models [29, 30]. Coinfections of more than one disease are also possible and their analysis requires more elaborate models. This is a real possibility with HIV and tuberculosis [96, 119, 133, 135, 140, 144, 154].

16.3 Antibiotic Resistance In short-term disease outbreaks, antiviral treatment is one of the methods used to treat illness and also to decrease the basic reproduction number R0 and thus to lessen the number of cases of disease. However, many infectious pathogens can evolve and generate successor strains that confer drug resistance [55]. The evolution of resistance is generally associated with impaired transmission fitness compared to the sensitive strains of the infectious pathogen [112]. In the absence of treatment, resistant strains may be competitively disadvantaged compared to the

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sensitive strains and may go extinct. However, treatment prevents the growth and spread of sensitive strains, and therefore induces a selective pressure that favors the resistant strain to replicate and restores its fitness to a level suitable for successful transmission [2]. This phenomenon has been observed in several infectious diseases, in particular for management of influenza infection using antiviral drugs [138]. Previous models of influenza epidemics and pandemics have investigated strategies for antiviral treatment in order to reduce the epidemic final size (the total number of infections throughout the epidemic), while preventing widespread drug resistance in the population [77, 107, 111, 113, 114]. Through computer simulations, these studies have shown that, when resistance is highly transmissible, there may be situations in which increasing the treatment rate may do more harm than good by causing a larger number of resistant cases than the decrease in cases produced by treatment of sensitive infections. A recent epidemic model [156] has exhibited such behavior and suggested that there may be an optimal treatment rate for minimizing the final size [107, 111, 114]. In diseases such as tuberculosis, which operate on a very long time scale, the same problems arise but the modeling scenario is quite different. It is necessary to include demographic effects such as births and natural deaths in a model. This means that there may be an endemic equilibrium, and that the disease is always present in the population. Instead of studying the final size of an epidemic to measure the severity of a disease outbreak, it is more appropriate to consider the degree of prevalence of the disease in the population at endemic equilibrium as a measure of severity. For diseases such as tuberculosis, in which there are additional aspects such as reinfection, there may be additional difficulties caused by the possibility of backward bifurcations. The importance of understanding the dynamics of tuberculosis treatment suggests that this is a topic that should be pursued [60]. Antibiotic-resistant bacterial infections in hospitals are considered one of the biggest threat to public health. The British Chief Medical Officer, Dame Sally Davies, noted that “the problem of microbes becoming increasingly resistant to the most powerful drugs should be ranked alongside terrorism and climate change on the list of critical risks to the nation.” Yet while antibiotic use is rising, not least in agriculture for farmed animals and fish, resistance is steadily growing. The challenges posed by the persistence, evolution, and expansion of resistance to antimicrobials are critically important because the number of drugs is limited and no new ones have been created for three decades [2, 16, 38, 65, 99]. We are facing a global crisis in antibiotics, the result of rapidly evolving resistance among microbes responsible for common infections that threaten to turn them into untreatable diseases. Every antibiotic ever developed is at risk of becoming useless. Antimicrobial resistance is on the rise in Europe, and elsewhere in the world. Dr. Margaret Chan, Director General of the World Health Organization, while addressing a meeting of infectious disease experts in Copenhagen, noted that “A post-antibiotic era means, in effect, an end to modern medicine as we know it. Things as common as strep throat or a child’s scratched knee could once again kill.

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For patients infected with some drug-resistant pathogens, mortality has been shown to increase by around 50 per cent.”1 Strategies suggested to curb the development of resistant hospital-acquired infections include antimicrobial cycling and mixing, that is, models of antibiotic use that make use of two distinct classes of antibiotics that may distributed over different schedules with the goal of slowing down the evolution of resistance. Cycling alternates both classes of drugs over pre-specified periods of time while mixing distributes both drugs simultaneously at random, that is, roughly half of the physicians would prescribe the first drug class while the other half would prescribe the second class. If the goal is to slow down single class drug resistance then “mixing” is the answer [16] while if the goal is to minimize dual resistance (if such a possibility exists) then the best option is cycling [38]. Of course, there are other factors that may accelerate resistance (physicians’ compliance) or slow down resistance (quarantine and isolation). All the above questions may be addressed via the use of contagion models [38].

16.4 Mobility The Global Commons are continuously reshaped by the ability of an increasing proportion of the human population to live, move, or trade nearly anywhere. Therefore, understanding the patterns of interactions between humans, between humans and vectors, and the patterns of individuals’ movement, particularly those who are highly mobile, is critical in guiding public health responses to disease spread. In today’s world, hosts’ knowledge of information about risk, combined with the ability of public health officials to measure and properly communicate, in a timely manner, real or perceived information on disease risks, affects our ability to derail the spread of emergent and re-emergent diseases, at scales that make a difference. Simon Levin showed that understanding scale-dependent phenomena is intimately tied in to our understanding on how information at particular scales impact other scales. His four decade old seminal paper establishing the relationships between processes operating at different scales that highlighted how macroscopic features arise from microscopic processes open the door to the theoretical advances that have dominated the study of ecological and epidemiological systems [101]. Specifically, the theory of metapopulations, common to the study of the models in this book [104, 155], was used to establish the role that localized disturbances have had in maintaining biodiversity [103, 127]. Kareiva et al. observe that there is a multitude of frameworks to study the role of disturbance, noting that, “Models that deal with dispersal and spatially distributed populations are extraordinarily varied, partly because they employ three distinct characterizations of space: as

1 The

Independent, Friday 16 March 2012.

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‘islands’ (or ‘metapopulations’), as ‘stepping-stones’, or as a continuum” [88]. We choose to deal with mobility in Chap. 15 and this chapter uses a metapopulation approach [80, 93, 104], with populations that exist on discrete “patches” defined by some characteristic(s) (i.e., location, disease risk, water availability, etc.). As is customary, patches are connected by their ability to transfer relevant information among themselves, which, in the context of disease dynamics, is modeled by the ability of individuals to move between patches. Patches may be constructed (defined) by species (human and mosquito) with movement explicitly modeled via patch-specific residence times and under a framework that sees disease dynamics as the result of location-dependent interactions and location characteristic average risks of infection [17, 18]. We observed that “It is therefore important to identify and quantify the processes responsible for observed epidemiological macroscopic patterns: the result of individual interactions in changing social and ecological landscapes” [25]. In the rest of this chapter, we touch on some of the issues calling for the identification of an encompassing theoretical explanatory framework or frameworks. We do this by identifying some of the limitations of existing theory, in the context of particular epidemiological systems. The goal is fostering and re-energizing research that aims at disentangling the role of epidemiological and socioeconomic forces on disease dynamics. In short, epidemic models on social landscapes are better formulated as complex adaptive systems. Now the question becomes, “How does such a perspective help our understanding of epidemics and our ability to make informed adaptive decisions?” These are huge complex questions whose answers have engaged a large number of interdisciplinary and trans-disciplinary teams of researchers. What may be promising directions? In what follows, we discuss some of the modeling used to address some of the challenges and opportunities that we believe must be considered in the field of theoretical epidemiology.

16.4.1 A Lagrangian Approach to Modeling Mobility and Infectious Disease Dynamics The deleterious impact of the use of cordons sanitaires [58, 100] to limit the spread of Ebola in West Africa points to the importance of developing and implementing novel approaches that may ameliorate the impact of disease outbreaks in areas where timely response to the emergence of novel pathogens is not possible at this time. Disease risk is a function of the scale and the level of heterogeneity considered. Risk varies by countries and within a country by areas of localized poverty, or as a function of the availability and quality of sanitary/phytosanitary conditions, or as a result of access and the quality of health care, or variability on the levels of individual education, or as a result of engrained cultural practices and norms. Travel and trade, easily bypassing in today’s world the natural or cultural boundaries defined by many of factors just outlined, are now seen as engines that

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drive the spread of pests and pathogens across regional and global scales. Hence, the identification of explanatory frameworks that help to disentangle the role of epidemiological, socioeconomic, and cultural perspectives on disease dynamics becomes evident and necessary in the Global Commons. Further, since the work of Sir Ronald Ross over a century ago [141], efforts to develop a mathematical framework that allow us to tease out the role of various mechanisms on disease spread while enhancing our understanding of what may be the most effective measures to manage or eliminate a disease, the fields of mathematical and theoretical epidemiology have developed into rich and useful fields of their own. Their role in the development of public health policy and the study of disease evolution within hosts (immunology) and between populations and its relationship to the study of host–pathogen interactions within ecology or community ecology are now integral components of the education and training of theoreticians and practitioners alike [1, 12, 19–23, 26, 41, 53, 54, 59, 74, 76, 82, 102, 122, 157]. The use of (per capita) contact or activity rates in modeling the interactions between individuals, that is, who mixes with whom or who interacts with whom, has been the natural social dynamics currency used to model human-to-human or vectorto-human interactions in the context of the transmission dynamics of communicable diseases. The “physics or chemistry traditions” are used to model disease transmission as the result of the “collisions” between individuals (with different energy or activity levels) in different epidemiological states. Further, movement, typically modeled using a metapopulation approach, is seen as the relocation between patches of non-identifiable individuals. The scholarly and extensive review in [83] addresses this perspective within homogeneous and heterogeneous mixing (age-structured) population models (see also [30]). Weakening the assumption of homogeneous mixing via contacts in epidemiology has been addressed using network-based analyses that identify host contact patterns and clusters [13, 120, 121, 128] (and references therein with [121] offering an extensive review). Focusing on how each individual is connected within the population has been used to address the effects of host behavioral response on disease prevalence (see [67, 68, 110] for a review). Other approaches have included the effects of behavioral changes triggered by “fear” and/or awareness of disease [56, 66, 131, 134]. Although this stress-induced behavior may benefit public health efforts in some cases, it can also cause somewhat unpredictable outcomes [75]. However, the fact remains that our ability to determine (and hence define) what an effective contact is in the context of communicable diseases, that is, our ability to measure the average number of contacts that a typical patch resident has per unit of time and where, has been hampered by high levels of uncertainty. Therefore, when we ask, what is the average rate of contacts that an individual has while riding a packed subway in Japan or Mexico City, or what is the average rate of contacts that an individual has at a religious event involving hundreds of thousands of people, including pilgrimages, one quickly arrives at the conclusion that different observers are extremely likely to arrive at very distinct understandings and quantifications of the frequency, intensity, and levels of heterogeneity involved. In short, this perspective puts emphasis on the use of a different currency (residence

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times) because measuring contacts at the places where the risk of infection is the highest, pilgrimages, massive religious ceremonies, “Woodstock time events”, packed subways, and other forms of mass gathering or transportation have not been done to the satisfaction of most researchers. The risk of acquiring an infectious disease within a flight can be measured at least in principle as a function of the time that each individual of x-type spends flying, the number of passengers, and the likelihood that an infectious individual is on board. For example, measuring the risk of acquiring tuberculosis, an airborne disease that may spread by air circulation in a flight, may be more a function of the duration of the flight and the seating arrangement than the average rate of contacts per passenger within the flight (see [31] and references therein). Furthermore, replication studies that measure risk of infection in a given environment may indeed be possible under a residence time model. In short, the risks of acquiring an infection can be quantified as a function of the time spent (residence time) within each particular environment. The Lagrangian modeling approach builds (epidemiological) models by tracking individuals’ patchresidence times and estimating their contacts according to the time spent in each environment [32]. The value of these models increases when we have the ability to assess risk as a patch-specific characteristic. In short, the use of a Lagrangian modeling perspective rather than the use of contacts is tied to the difficulties that must be faced when the goal is to measure the average rate of contacts per type-x individual in the environments that facilitate transmission the most. The Lagrangian approach is highlighted here via the formulation of a disease model involving the joint dynamics of an n-patch geographically structured population with individuals moving back and forth from their place of residence to other patches. Each of these patches (or environments) is defined by its associated risk of residence-time infection. Patch risk measurements account for environmental, health, and socioeconomic conditions. The Lagrangian approach [73, 125, 126] keeps track of the identity of hosts regardless of their geographical/spatial position. The use of Lagrangian modeling in living systems was, to the best of our knowledge, pioneered and popularized by Okubo and Levin [125, 126] in the context of animal aggregation. Recently, Lagrangian approaches have also been used to model human crowd movement and behavior [15, 49, 78, 79] and in the context of bioterrorism [31]. Here, host-residence status and mobility across patches are monitored with the help of a residence times matrix P = (pij )1≤i,j ≤n , where pij is the proportion of time residents of Patch i spend in Patch j . Letting Ni denote the population of Patch i predispersal, that is, when patches are isolated, we conclude that effective

population size in Patch i, at time t, is given by nj=1 pj i Nj . That is, the effective population within each patch must account for the residents and visitors to Patch i at time t. A typical SIS model captures this Lagrangian approach in an n - patch setting via the system of nonlinear differential equations:

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S˙i = bi − di Si + γi Ii −

n (Si infected in Patch j ) j =1

(16.1)

n (Si infected in Patch j ) − γi Ii − di Ii , I˙i = j =1

where bi , di , and γi denote the constant recruitment, the per capita natural death, and recovery rates, respectively, in Patch i. The effective population nj=1 pij Nj in

each Patch i, i = 1, . . . , n includes nj=1 pij Ij infected individuals. Therefore, the infection term is modeled as follows:

n pkj Ik . Si infected in patch j = βj × pij Si × nk=1 k=1 pkj Nk

The likelihood of infection in each patch is tied to the environmental risks, defined by the “transmission/risk” vector B = (β1 , β2 , . . . , βn )t and the proportion of time spent in particular area. Letting I = (I1 , I2 , . . . , In )t , N¯ = ¯ d = (d1 , d2 , . . . , dn )t , and γ = (γ1 , γ2 , . . . , γn )t ( bd11 , bd22 , . . . , dbnn )t , N˜ = Pt N, allows to rewrite System 16.1 in the following single vectorial form I˙ = diag(N¯ − I )Pdiag(B)diag(N˜ )−1 Pt I − diag(d + γ )I.

(16.2)

The dynamics of the disease in all of the patches depends on the patch connectivity structure. Therefore, if the residence-time matrix P is irreducible, patches are strongly connected, then system 2 supports a sharp threshold property. That is, the disease dies out or persists (in all patches) whenever the basic reproduction number R0 is less than or greater than unity [18]. R0 is given by R0 = ρ(diag(N¯ )Pdiag(B)diag(N˜ )−1 Pt V −1 ), where ρ denotes the spectral radius and V = −diag(d + γ ). The dynamics of the system when the matrix P is not irreducible can be characterized using the following patch-specific basic reproduction numbers: βi R0i (P) = × γi + di

 n  βj j =1

βi

⎛

pij

pij ⎝ n

  bi di

k=1 pkj bk dk

⎞

⎠.

The disease persists in Patch i whenever R0i (P) > 1, whereas the disease dies out in Patch i if pkj = 0 for all k = 1, . . . , n, and k = i,provided pij > 0 and R0i (P) < 1.

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Patch-specific disease persistence can be established using the average Lyapunov theorem [86] (see [18] for more details). In Model 16.2, human behavior is crudely incorporated through the use of a constant mobility matrix P. The role that adaptive human behavior may play in response to disease dynamics is captured, also rather crudely, via a phenomenological approach that assumes that individuals avoid or spend less time in areas of high prevalence. This effect is captured by placing natural restrictions on the entries of p (I ,I ) p (I ,I ) P. The inequalities ij ∂Iij j ≤ 0 and ij ∂Iii j ≥ 0, for (i, j ) ∈ 1, 2, guarantee the expected behavioral response. An example of such dependency could be captured σ +σ I +I 1+Ii by the following functions: pii (Ii , Ij ) = ii1+Iiii +Ii j j and pij (I1 , Ij ) = σij 1+I , i +Ij

2 for (i, j ) ∈ 1, 2 and σij = pij (0, 0), are such that j =1 σij = 1. The simulation below shows how a crude, density-dependent modeling mobility approach can alter the expected disease dynamics from those generated under constant P (Figs. 16.1 and 16.2). In the special case, where there is no movement between patches (p12 = p21 = σ12 = σ 21 = 0), that is, there is no behavioral change, the two populations support, as expected, the same dynamics (see the blue curves in Figs. 16.1 and 16.2). The speed at which the vector-borne Zika virus disease spread throughout Latin America, Central America, and the Caribbean (then hitting Mexico and the United States) was strongly linked to human mobility patterns. Travelers transport the disease and infect native mosquitoes. Here, it is assumed that vector mobility is negligible and the assumptions proceed to incorporate the life history and epidemiology of mosquitoes [10, 84, 98, 108, 109, 141], which can be effectively captured

Fig. 16.1 Dynamics of the disease in Patch 1 for three special cases. The symmetric residence times (p12 = p21 = σ12 = σ21 = 0.5) are described by the solid and dashed black curves. The blue curves represent the case where there is no movement between patches, that is, p12 = p21 = σ12 = σ21 = 0. The red curves represent the high-mobility case for which p12 = p21 = σ12 = σ21 = 1. If there is no movement between the patches (blue curves), the disease dies out in the low risk Patch 1 = 0.7636. The vertical axis represents the prevalence of 1 in both approaches with R01 = d1β+γ 1 the disease in Patch 1. Figure courtesy of Ref. [18]

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Fig. 16.2 Dynamics of the disease in Patch 2. In the high-mobility case p12 = p21 = σ12 = σ21 = 1 (and then p11 = p22 = σ11 = σ22 = 0), the disease dies out (solid red curve) for P constant, with 1 = 0.8571. For the constant residence-time matrix, the system is strangely decoupled R˜02 = γ2β+d 2 because individuals of Patch 1 spend all their time in Patch 2, whereas individuals of Patch 2 spend all their time in Patch 1. Hence, Patch 2 individuals (d2 and μ2 ) are subject exclusively to the environmental conditions that define Patch 1 (β1 ), and so the basic reproduction of the “isolated” 1 Patch 1 is R˜02 = γ2β+d and the disease dies out because R˜02 = 0.8571. The disease persists if P 2 state-dependent (dashed red curve) as p12 (I1 , I2 ) = I2 1+I1 +I2 ,

and p22 (I1 , I2 ) =

I1 1+I1 +I2 .

1+I1 1+I1 +I2 , p21 (I1 , I2 )

Figure courtesy of Ref. [18]

=

1+I2 1+I1 +I2 ,p11 (I1 , I2 )

=

by decoupling host and vector mobility [98, 145]. Figure 16.3 and System 16.3 illustrate the approach. A Lagrangian model based on residence times has been proposed recently for vector-borne diseases like dengue, malaria, and Zika [17]. The appropriateness of the Lagrangian approach for the study of the dynamics of vector-borne diseases lies also in its assessment of the life-history specifics of the vector involved [145]. I˙h = βvh diag(Nh − Ih )Pdiag(a)diag(Pt Nh )−1 Iv − diag(μ + γ )Ih I˙v = βhv diag(a)diag(Nv − Iv )diag(Pt Nh )−1 Pt Ih − diag(μv + δ)Iv . (16.3) Lagrangian approaches have been used to model vector-borne diseases (see [48, 87, 139, 142, 153] and other references contained therein), although these researchers have not considered the impact that the residence-time matrix P may have on patch effective population size. Specifically, in [48, 142], the effects of movement on patch population size at time t are ignored, namely, the population size in each Patch j is fixed at Nj . In [139], it is assumed that human mobility across patches does not produce any “net” change on the patch population size. On the other hand, in Model 16.3 the relationship between each patch population and mobility is dynamic and explicitly formulated. Moreover, the limited (vector

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Fig. 16.3 Flow diagram of a Lagrangian model in which the host structure is decoupled from the vectors’ structure. Figure courtesy of Ref. [17]

mobility is ignored) Lagrangian approach used here to model the dynamics of vector-borne diseases captures some unique features because the “spatial” structure of mosquitoes is not the same as that of humans. Mosquitoes are stratified into m patches (that may represent, for example, oviposition or breeding sites or forests) with infection taking place still within each Patch j , characterized by its particular risk βvh aj for j = 1, . . . , m. Here, βvh represents the infectiousness of human to mosquitoes bite with aj denoting the per capita biting rate in Patch j . Hosts, on the other hand, are structured by social groups or age classes (n). This residence habitat division can be particularly useful in the study of the impact of target control strategies. The model in [17] describes the interactions of n host groups in m patches via System 16.3, where Ih = [Ih,1 , Ih,2 , . . . , Ih,n ]t , Iv = [Iv,1 , Iv,2 , . . . , Iv,m ]t

Nh = [Nh,1 , Nh,2 , . . . , Nh,n ]t , N¯ v = [N¯ v,1 , N¯ v,2 , . . . , N¯ v,m ]t

δ = [δ1 , δ2 , . . . , δm ]t , a = [a1 , a2 , . . . , am ]t , andμ = [μ1 , μ2 , . . . , μn ]t .

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The infected host population is denoted by the vector Ih and the host population by Nh . The infected vector population is denoted by Iv and the mosquito population by Nv . The parameters ai , δi , and μv denote the biting, death rate of control, and natural death rate of mosquitoes in Patch j , for j = 1, . . . , m. The infectiousness of human to mosquitoes is βvh , whereas the infectiousness of mosquitoes to humans is given by βh v. The host recovery and natural mortality rates are given, respectively, by γ and μ. Finally, the matrix P represents the proportion of time host of group i, i = 1, . . . , n, spend in Patch j , j = 1, . . . , m. The basic reproduction number of Model 3, with m patches and n groups, is given by R02 (m, n) = ρ(Mvh Mhv ), where Mhv = βhv diag(a)diag(Pt Nh )−1 diag(Nv )Pt diag(μ + γ )−1 and Mvh = βvh diag(Nh)Pdiag(Pt Nh )−1 diag(a)diag(μv + δ)−1 . If the host–vector network configuration is irreducible, then System 16.3 is cooperative and strongly concave with an irreducible Jacobian, hence the theory of monotone systems, particularly Smith’s results [146], guarantee the existence of a sharp threshold. That is, the disease-free equilibrium is globally asymptotically stable if R02 (m, n) is less than unity and a unique globally asymptotic stable interior endemic equilibrium exists otherwise. The effects of various forms of heterogeneity on the basic reproduction number have been explored in [17], and we have found, for example, that the irreducibility of the residence-time matrix P is no longer sufficient to ensure a sharp threshold property, although the irreducibility of the host–vector network configuration is necessary for such property [17]. The Lagrangian approach to disease modeling can use contacts [32] or times or both as its currency. Here, we choose time-spatial-dependent risk, that is, we choose to handle social heterogeneity by keeping track of individuals’ social or geographical membership. In this context, it is possible to include adaptive responses, for example, via the inclusion of prevalence-dependent dispersal coefficients. In this setting, the underlying hypothesis is that host behavioral responses to disease are automatic: either constant or following a predefined function. The average residence time P incorporates the average behavior of all hosts in each patch. This assumption is rather crude because it implicitly assumes that hosts have accurate information on health status and patch prevalence and respond to risk of infection accordingly. The incorporation of the role that human decisions, as a function of what individuals value and the cost that individuals place on these choices and trade-offs, within systems that account for the overall population disease dynamics has been addressed recently [61, 132] and discussed in economic epidemiology.

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16.5 Behavior, Economic Epidemiology, and Mobility The movement/behavior of individuals within and between patches may be driven by real or perceived personal economic risk and associated social dynamics. Embedding behavioral-driven decisions in epidemiological models has shed new perspectives on the modeling of disease dynamics [61], expanding available option to manage infectious diseases [44, 62]. Economic epidemiological modeling (EEM) has a history of addressing the role of individuals’ behavior when facing the risk of disease. However, it has often failed to incorporate within host-pathogen feedback mechanisms [34–36, 52, 70, 97, 137]. EEMs that account for host-pathogen feedback mechanisms has propelled the study of the ways that contact decisions impact disease emergence or alter infectious disease-transmission dynamics. Decisions involved may include the determination to engage in trade on particular routes [89, 94, 95, 129], or to travel to specific places [62, 147–149], or to make contact with or to avoid particular types of people [61, 63, 116]. EEMs advance the view that the emergence of novel zoonotic diseases, such as SARS or the Nipah virus, depend on the choices that bring people into contact with other species [50, 51]. EEMs are usually built under the assumption that associated disease risks are among the factors that individuals must consider when making decisions. Individual decision-making processes, within epidemic outbreaks, must incorporate the humans’ cost–benefit-driven adaptive responses to risk.

16.5.1 Economic Epidemiology Simple EEMs are, by mathematical necessity, initially built on classical compartmental epidemiological models that account for the orderly transition of individuals facing a communicable disease, through the susceptible, infected, and recovered disease stages: the result of social and environmental interactions. EEMs assume that the amount of activity one participates in, with whom, and where may all be envisioned as the solutions to an individual decision problem. It is assumed that individual decision problems are generated by rational-value formulations based on (driven by) personal, real or perceived, cost of disease, and disease avoidance: decisions constrained by underlying population-level disease dynamics. Thus, finding effective ways of modeling rational value connections to individualized cost–benefit analyses of disease risk is fundamental to the building of useful EEMs. It is a quite challenging enterprise. EEM approaches have precursors in the epidemiological literature [9, 64, 69]. EEM construction has been strongly influenced by past and ongoing work on the exploitation of species [45–47], a literature that addresses optimal harvesting questions in the context of wild species, or the control of invasive pests, or the management of forestry system. The methodology for modeling behavior within an EEM rests on a proper specification of behavioral costs and a description of

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the payoffs linked to such behaviors; the stipulation of an appropriate objective function, congruent with the decision-makers’ goals; the coupling to the dynamics of the natural resource and/or infectious human capital; and the mechanisms available for a decision-maker to alter his or her behavior and the behaviors of those around him or her. Although not all motivations for mitigation against infection are monetary in nature, we choose to refer to them as economic. Modeling whether or not an individual undertakes infection-causing behavior provides a natural starting point since it is connected to the rate of generation of secondary cases of infection per unit of time, the so-called incidence rate. A simple incidence function that captures the instantaneous expectation of the rate of new infections at a given time is therefore given by S(t)cPSI (t)ρ, where S(t) is the number of individuals susceptible to the disease, c is the average amount of activity they engage in, PSI (t) is the probability that a unit of such activity takes the susceptible individual in contact with infectious individuals/material, and ρ is the probability that such contact successfully infects. A decision to reduce the volume of activity one engages in (lowering c) has been shown in many cases to be phenomenologically identical to reducing one’s chances of coming in contact with infection (lowering PSI (t)) by altering where the activity takes place and with whom one engages or by substituting a particular behavior for a riskier one [62, 118]. The modeling assumes that individuals derive benefits from making contacts but may incur costs associated with an infection. Hence, the modeling assumes that activity volume or contacts are chosen to maximize expected utility (rudimentarily, benefit less cost), balancing the marginal value of a contact against the increased risk of infection. The utility function is assumed to depend on the health status of the individual and the contacts that they make, that is, the utility of a representative individual of health status h is given, for example, by the function Uh = U (h, C h ).

(16.4)

The utility function is assumed to be concave, decreasing in illness and increasing in contacts. If the probability of transitioning from susceptible to infected health status depends on the rate of contacts, the optimal choice of contacts is the solution to a dynamic programming problem:

Vt (h) = max Cs

⎧ ⎨ ⎩

Ut (ht , Cth ) + r

j

⎫ ⎬

ρ hj Vt+1 (j ) , ⎭

(16.5)

where r is the discount rate and ρ hj is the probability of transition from health state h to health state j . This probability depends on the current state of the system, {S(t), I (t), R(t)}, the behavior of individuals in other health classes, C −h , and

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the behavior of individuals in the decision-makers’ own health class, C¯ h . In short, we have a complex adaptive system where individuals within the model, in this example, impact disease outcomes (through changes in the incidence). Eqs. (16.4) and (16.5) are both optimized from an individual perspective. Within this individual context, EEMs have shown that individual distancing, conditional on health status, plays an important role in the spread of infectious disease. However, it has also been shown that the provision of incentives for infectious individuals to self-quarantine is likely to be welfare-enhancing [43, 44, 61, 115]. Thus, understanding how the individual responds to relative costs of disease and disease prevention is critical to the design of public policy that affects those costs. Indeed, the role of recovered individuals in protecting susceptible individuals has been generally overlooked in public health interventions, and yet it is known that their behavior is, in fact, critical to disease management due to the positive externality the individuals’ contacts generate once in an immune, non-disease-transmitting state [63]. The benefits of herd immunity include the positive externality associated with acquired immunity but may, in turn, be nullified by nontargeted social-distancing policies that induce such immune individuals to reduce contacts. By incentivizing the maintenance of contacts by recovered individuals policy may lower the probability of susceptible individuals contacting infected individuals and/or allow susceptible and infected individuals to individually increase contacts without changing the probability of infection.

16.5.2 Lagrangian and Economic Epidemiology Models Theoretical epidemiology aims to disentangle the role of epidemiological and socioeconomic forces on disease dynamics. However, the role of behavior and individual decisions in response to a changing epidemic landscape has not been tackled systematically. In this chapter, we highlight alternative ways for modeling disease transmission that can use contacts as its currency or residence times or both. It seems evident that the use of contacts, in the context of influenza, Ebola, tuberculosis, or other communicable diseases (as opposed to sexually transmitted diseases), while intellectually satisfying, fails to recognized the fact that contacts cannot be measured effectively in settings where the risk of acquiring such infections is the highest. In fact, when contact-based models are fitted to data, it has become clear that contact rates play primarily the role of fitting parameters; in other words, if the goal is connecting models to data that include transmission mechanisms, then the use of contacts has serious shortcomings. Therefore, in order to advance the role of theory, we need models that are informed by data. Hence, the need to invest on efforts that bring forth alternative modes of modeling. While Lagrangian approaches are not a panacea, their use extends the possibilities because they depend on parameters like residence times and average time to infection for a given environment (risk), that is, parameters that can be measured. Frameworks should be explored and compared

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and their analyses contrasted. We have revisited recent work that equates behavior with cost–benefit decisions, which, in turn, are linked, within our framework, to health status and population-level dynamics, the components of a complex adaptive system. Connecting the Lagrangian movement-modeling approach with EEMs seems promising, albeit computationally and mathematically challenging. However, as discussed in [117], the perception that the benefits of disease control are limited by the capacity of the weakest link in the chain to respond effectively is not a basic result of EEM models, which actually show that it may not be in within the ability of an individual in a poor community/country to do more risk mitigation. The need for richer communities or nations to find ways to incentivize greater levels of diseaserisk mitigation in poor countries may be the best approach. Simon Levin, in his address as the 2004 recipient of the Heineken award, placed our narrow perspective in a broader powerful context: A great challenge before us is thus to understand the dynamics of social norms, how they arise, how they spread, how they are sustained and how they change. Models of these dynamics have many of the same features as models of epidemic spread, no great surprise, since many aspects of culture have the characteristics of being social diseases. 1998 Heineken award winner Paul Ehrlich and I have been directing our collective energies to this problem, convinced that it is as important to understand the dynamics of the social systems in which we live as it is to understand the ecological systems themselves. Understanding the links between individual behavior and societal consequences, and characterizing the networks of interaction and influence, create the potential to change the reward structures so that the social costs of individual actions are brought down to the level of individual payoffs. It is a daunting task, both because of the amount we still must learn, and because of the ethical dilemmas that are implicit in any form of social engineering. But it is a task from which we cannot shrink, lest we squander the last of our diminishing resources.

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Epilogue

This book, inspired by our textbook Mathematical Models in Population Biology and Epidemiology [4], attempts to draw attention to the fields of computational, mathematical, and theoretical epidemiology (CMTE) with the goal of reaching researchers and practitioners whose interests and questions overlap with those posed by individuals or groups working in the fields of epidemiology, ecology, population biology, public health, and applied mathematics. Different variants of contagion models are used to study the transmission dynamics of communicable and vectorborne diseases. Contagion models are now being used extensively in the study of the dynamics of socio-economic-epidemiological processes [3, 11, 16]. One of the objectives of this book is to highlight the commonality of modeling approaches within the fields of epidemiology and the population dynamics of host– pathogen or host–parasite systems and consequently the intimate mathematical connections between ecology, epidemiology, and population biology. The mathematical focus is primarily on the applications of CMTE to the study of disease dynamics in large populations and, consequently, the book is dominated by the use of deterministic models. We tacitly recognize the challenges posed by the socio-behavioral dynamics that underline the world where disease appears, expands, persists, and evolves. Researchers and practitioners willing to immerse themselves in the topics addressed will value the importance and relevance of building research programs at the interface of CMTE and public health, guided by adopting a holistic perspective. The challenges faced by a world shaped and re-shaped by mobility, trade, changing demographics, war, violence, and health and economic disparities highlight the importance of training researchers at the intersection of CMTE and public health. The contents of this volume have been shaped by the understanding gained from the study of models for specific diseases and their analyses as well as by the challenges of connecting and simplifying processes that impact disease dynamics across scales and levels of organization, as is the case, for example, of immunology and epidemiology.

© Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9

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Our systems level thinking was instigated by the 1992 NSF workshop Mathematics and Biology: the Interface Challenges and Opportunities [13]. The workshop and the meeting’s report focused on the interface between mathematics and biology, which “. . . has long been a rich area of research, with mutual benefit to each supporting discipline.” Classical areas of investigation “such as population genetics, ecology, neurobiology, and 3-D reconstructions, have flourished [over] the past twenty years.” Interactions between mathematicians and biologists have changed dramatically, reaching out to encompass areas of both biology and mathematics that previously had not benefited from the closer integration of theory and experiment due to the increased reliance on high-speed computation. The report documents the participants’ enthusiastic conclusion that mathematical and computational approaches are essential to the future of biology and that biological applications will continue to contribute to the vitality of mathematics, as they have since the days of Vito Volterra. The ideas and vision expressed in this nearly three-decades old report have been expanded, reformulated, and enriched to the point that three decades later it has become routine to hold specialized workshops in mathematical epidemiology, mathematical immunology, mathematical physiology, or mathematical ecology, to name but a few. In 2009, most of the NSF directorates supported the workshop Towards a Science of Sustainability, an effort to bring to the forefront of NSF the importance of convergence research. The report Toward a Science of Sustainability [9] states that “Building a science of sustainability. . . requires a truly multi-disciplinary approach that integrates practical experience with knowledge and know-how drawn from across the natural and social sciences, medicine and engineering, and mathematics and computation.” This National Science Foundation (NSF) report, carried out with the full support of most Directorates at the Foundation, places in context the importance of the challenges and opportunities posed by sustainability science. The importance of sustainability thinking in epidemiology and public health has been central to their goal of reducing or ameliorating the impact of disease from the individual to the population level and beyond. The need to manage antibiotic resistance, or to formulate and implement sustainable health solutions, or the effective management of vector-borne diseases, or working on identifying ways of assessing the role of travel and trade and their impact on disease dynamics, or the challenges posed by the ecology of infectious disease, or the efforts to reduce the prevalence or stop the expansion of health disparities would all benefit from what is being learned within the science of sustainability [14]. We believe that the chapters of this book provide a good start for those interested not only on epidemiological applications but also on processes that can be modeled as contagion processes as the surge in applications in the context of the behavioral, economic, and social sciences demonstrate [3, 11]. The possible deliberate release of biological agents and their consequences has moved to the forefront of the concerns of the Global Commons. Its potential impact is being addressed in part by the models and methods developed over the past century within the field of computational and mathematical epidemiology.

Epilogue

535

Specific models have been used to address and assess the role of disease and interventions on human or animal populations or agricultural systems. Studies on the potential impact of the deliberate release of biological agents have instigated the development of mathematical frameworks capable of addressing the impact of foot and mouth disease, or the role of mobility and trade, and particularly situations when a released agent is aimed at unsuspecting populations. In the Report generated at a 2002 DIMACS working group meeting, Mathematical Sciences Methods for the Study of Deliberate Releases of Biological Agents and their Consequences [5] these issues were addressed soon after the events of September 11 of 2001. The Centers for Disease Control (CDC) “. . . in the early 1950s established and developed intelligence epidemiological services. This decision, driven in part by national concerns about the potential use of biological agents as a source of terror, was one of the first systematic responses to bioterrorism . . . The impact of deliberate releases of biological agents (Foot and Mouth, Mediterranean Fruit Flies, Citrus rust, etc.) on agricultural systems and/or our food supply needs to be addressed and evaluated. For example, foot and mouth disease was most likely accidentally introduced in Britain nearly simultaneously at multiple sites via the cattle food supply and agricultural personnel movement. Hence, it was difficult to contain this outbreak despite Britain’s effective post-detection response (stamping-out). The costs associated with its containment have been estimated to be over 15 billion dollars. The use of agents like anthrax highlights the need to look at existing models for the dispersion of pathogens in buildings (models of air-flow in buildings) and in water systems (e.g. dispersion while flowing through pipes) . . . new paradigms are needed for the study of releases of these agents in rather unconventional ways.” The challenges posed by the Global Commons have reached critical stages as travel, trade, immigration, population growth, conflicts, disease and changing economic and demographic factors continuously exhibit our inability to reduce the levels of uncertainty when assessing the impact of disease within the socio-epi sphere that we live in. The opportunities given to modelers, epidemiologists, and public health experts to improve health outcomes using mathematical modeling have re-energized the intersecting fields of computational, mathematical, and theoretical epidemiology or CMTE just as the world’s population surpasses the seven billion mark. Further, the fields of data science, demography, social dynamics, public health policy, behavioral sciences, modeling, computer science and applied mathematics continue to enhance our understanding of problems that can be better understood and managed from what can be learned from the study of specific questions that can be often addressed through synergistic approaches involving the mathematical, health, and behavioral sciences. Efforts to advance relevant theoretical work while searching for solutions to pressing epidemiological questions is becoming the new standard, some would say the norm. Hence, grounding ourselves within the context of human managed or operated systems is not only critical but also highlights the importance of the following question: How can decision-makers build and implement robust policies anchored on the best information and knowledge available given that social landscapes evolve and change as a function of the

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Epilogue

decisions that individuals make every single moment in response to real or perceived information? The challenges of the twenty-first century have brought the role of individual and group decisions to the forefront of most efforts aimed at ameliorating the impact and consequences of emergent, re-emergent, and endemic diseases. The technical advances of the twenty-first century have put the power of the quantitative sciences in the hands of policy and decision makers. Progress depends on the researchers’ ability to address disease dynamics, evolution and epidemiological management challenges tied in to current and changing understanding of disease processes across various levels of organization within systems often operating simultaneously over fast, intermediate, and slow time scales. The need to model and quantify uncertainty within multi-scale, multi-state human-disease systems increases the role that the impact of human behavior and decisions play in altering the social and biological landscapes on which diseases operate. Disentangling the complications of disease dynamics, control, and individuals’ decisions demand the use of a holistic perspective since, in fact, most often we are dealing with complex adaptive systems [12]. Simon Levin put it best when he observed (in his address as the 2004 recipient of the Heineken award), “A great challenge before us is thus to understand the dynamics of social norms, how they arise, how they spread, how they are sustained and how they change. Models of these dynamics have many of the same features as models of epidemic spread, no great surprise, since many aspects of culture have the characteristics of being social diseases. The 1998 Heineken award winner Paul Ehrlich and I have been directing our collective energies to this problem, convinced that it is as important to understand the dynamics of the social systems in which we live as it is to understand the ecological systems themselves. Understanding the links between individual behavior and societal consequences, and characterizing the networks of interaction and influence, create the potential to change the reward structures so that the social costs of individual actions are brought down to the level of individual payoffs. It is a daunting task, both because of the amount we still must learn, and because of the ethical dilemmas that are implicit in any form of social engineering. But it is a task from which we cannot shrink, lest we squander the last of our diminishing resources.” The challenges posed by the explosive growth of antibiotic resistance are immense. The head of the WHO while addressing a meeting of infectious disease experts in Copenhagen highlighted recently the global crisis in antibiotics, the result of “rapidly evolving resistance among microbes responsible for common infections that threaten to turn them into untreatable diseases every antibiotic ever developed was at risk of becoming useless.” No new classes of antibiotics have been developed since 1987. Professor Nigel Brown, president of the Society for General Microbiology remarks that immediate action by scientists is required if we are going to identify and mass produce new antibiotics; the kind of effort needed to tackle the problem of antimicrobial resistance and its transmission, particularly in the context of nosocomial (in hospital) infections.

References

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The challenges and opportunities facing the fields of epidemiology and public health are timely, pressing, and complex. The contributions of the fields of computational, mathematical, and theoretical epidemiology continue to expand often responding immediately epidemiological emergencies [2, 6, 7]. Bill Gates, in Business Insider has recently warned the world (April 27, 2018, and May 12, 2018) about the consequences of its lack of preparedness for the sureto-be coming influenza pandemic. This book, through the study of communicable and vector-borne diseases, some re-emergent, and some emergent diseases, provides a solid training on modeling approaches and the mathematical methods that may be useful in the formulation and analyzes of population-level modeling efforts aimed at containing or ameliorating the impact of childhood diseases, sexually-transmitted diseases, HIV, tuberculosis, influenza, vector-borne diseases, and more. We have included applications that account for the impact of host or pathogen heterogeneity on disease dynamics and control. We have made efforts to enhance the modeling tool kit by outlining and comparing the use of “Lagrangian” and “Eulerian” modeling perspectives. The issue of time scales has been addressed in various forms throughout including via the clear differentiating that we have made of the differences that come from the study and modeling of short and long-term dynamics as well as on the consequences of including population structure [15]. Some of the limitations found in this volume are tied in to our deliberate omission of stochastic models [1] or at least some of the specifics of parameter estimation [8]. Further, we have not included the role of agent based modeling [10] or many of the advances and perspectives that network theory has brought into epidemiology. This book captures our biased perspectives on the role of computational, mathematical, and theoretical epidemiology in the study of disease dynamics and public health. However, it does not provide a comprehensive view of CMTE, just the basics. We believe nevertheless that its content would be useful and relevant to those interested in addressing the challenges posed by disease dynamics and control.

References 1. Allen, L. (2010) An Introduction to Stochastic Processes with Applications to Biology, CRC Press. 2. Arino, J., F. Brauer, P. van den Driessche, J. Watmough, J. Wu (2006) Simple models for containment of a pandemic, J. Roy. Soc. Interface, https://doi.org/10.1098/rsif.2006.0112. 3. Bettencourt, L.M., A. Cintrón-Arias, D. J. Kaiser, and C. Castillo-Chavez (2006) The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models, Physica A: Statistical Mechanics and its Applications. 364: 513-536. 4. Brauer F. and C. Castillo-Chavez (2012) Mathematical models in population biology and epidemiology. No. 40 in Texts in applied mathematics, New York: Springer, 2nd ed. 2012. 5. Castillo-Chavez, C. and F. Roberts (2002) Report on DIMACS working group meeting: Mathematical sciences methods for the study of deliberate releases of biological agents and their consequences. Biometrics Unit Technical Report (BU-1600-M) Cornell University

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6. Chowell, D., C. Castillo-Chavez, S. Krishna, X. Qiu, and K.S. Anderson (2015) Modelling the effect of early detection of Ebola. The Lancet Infectious Diseases, 15: 148–149. 7. Chowell, G., P.W. Fenimore, M.A. Castillo-Garsow, and C. Castillo-Chavez (2003) SARS outbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation as a control mechanism. J. Theor. Biol. 224: 1–8. 8. Chowell, G., J. M. Hayman, L. M. A. Bettencourt, C. Castillo-Chavez (2014) Mathematical and Statistical Estimation Approaches in Epidemiology, Springer. 9. Clark W.C., S.A. Levin (2010) Toward a science of sustainability: Executive, Levin SA, Clark WC (Eds.), Report from Toward a Science of Sustainability Conference, Airlie Center, Warrenton, Virginia. 10. Epstein, J. and S. Epstein (2007) Generative Social Science: Studies in Agent-Based Computational Modeling, Princeton University Press. 11. Fenichel, E. P., Castillo-Chavez, C., Ceddia, M. G., Chowell, G., Parra, P. A. G., Hickling, G. J., . . . & Springborn, M. (2011 Adaptive human behavior in epidemiological models. Proc. Nat. Acad. Sci. textbf108: 6306–6311. 12. Levin, S. (1999) Fragile Dominion: Complexity and the Commons. Perseus Publishing, Cambridge, Massachusetts. 13. Levin, S. (1992) Mathematics and Biology: The Interface, Challenges and Opportunities (Ed. Simon A Levin). 14. Muneepeerakul, R., & Castillo-Chavez, C. (2015) Towards a Quantitative Science of Sustainability. 15. Song, B., C. Castillo-Chavez, J. P. Aparicio (2002) Tuberculosis models with fast and slow dynamics: the role of close and casual contacts, 180, 187–205. 16. Towers, S., A. Gomez-Lievano, M. Khan, A. Mubayi, C. Castillo-Chavez (2015) Contagion in Mass Killings and School Shootings, PLoS One, https://doi.org/10.1371/journal.pone. 0117259.

Appendix A

Some Properties of Vectors and Matrices

A.1 Introduction A single linear algebraic equation in one unknown has the form Ax = b. A system of m linear equations in n variables is a system of equations of the form a11 x1 + a12 x2 + · · · + a1n xn = b1 a21 x1 + a22 x2 + · · · + a2n xn = b2 ............... an1 x1 + an2 x2 + · · · + ann xn = bn The language of vectors and matrices makes it possible to write this system in the simple form Ax = b, where A is an m × n matrix and b is a column vector. We will develop the elementary theory of vectors and matrices to show how to use this language to simplify the analysis of linear systems. This will be useful in the study of systems of differential equations that arise in epidemic models. In models consisting of a system of two differential equations in two unknown functions the matrices and vectors involved will have m = n = 2, and readers should interpret the results in this chapter with m = n = 2. We may think of matrix algebra as machinery that will allow us to simplify the language of epidemiological models.

© Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9

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A Some Properties of Vectors and Matrices

A.2 Vectors and Matrices A real n-vector is an ordered n-tuple of real numbers of the form v = [a1 , a2 , . . . , an ]. We also write v in the form (column vector) ⎡ ⎤ a1 ⎢a2 ⎥ ⎢ ⎥ v = ⎢ . ⎥. ⎣ .. ⎦ an

(A.1)

In these notes, we use the form (A.1), since this will simplify calculations later. The real numbers will be called scalars. We have the following two vector operations: 1. Addition, which is given by the formula ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ a1 a1 + b1 b1 ⎢ .. ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎣ . ⎦ + ⎣ . ⎦ = ⎣ . ⎦; bn

an

an + an

2. Scalar multiplication, defined by ⎡ ⎤ ⎡ ⎤ a1 λa1 ⎢ ⎥ ⎢ ⎥ λ ⎣ ... ⎦ = ⎣ ... ⎦ . an

λan

We define the scalar product (also called inner or dot product) of two vectors by ⎡ ⎤ ⎡ ⎤ b1 a1 ⎢ .. ⎥ ⎢ .. ⎥ ⎣ . ⎦ · ⎣ . ⎦ = a1 b1 + . . . + an bn . bn

an

Example 1 Consider the vectors ⎡ ⎤ ⎤ −2 1 v1 = ⎣ 3 ⎦ , v2 = ⎣ 1 ⎦ . 2 −2 ⎡

A

Some Properties of Vectors and Matrices

541

Then ⎡ ⎤ −1 ⎣ v1 + v2 = 4 ⎦ , 0

⎡

⎤ 3 3v1 = ⎣ 9 ⎦ , −6

v1 · v2 = (1)(−2) + (3)(1) + (−2)(2) = −3. It is not hard to see that the operations defined above have the following properties: u+v = v+u (u + v) + w = u + (v + w) λ(u + v) = λu + λv u·v = v·u u · (v + w) = u · v + u · w u · (λv) = λu · v An m × n matrix is an array of mn real numbers with m rows and n columns: ⎡

a11 a12 ⎢ a21 a22 ⎢ A=⎢ . .. ⎣ .. . am1 am2

... ... .. .

⎤ a1n a2n ⎥ ⎥ .. ⎥ . . ⎦

(A.2)

. . . amn

The matrix A is also written as A = (aij ). The size of A is m × n. Matrices of the same size can be added, and matrices can be multiplied by scalars, according to the following rules: ⎡

a11 ⎢ . ⎢ . ⎣ . am1

... .. . ...

⎤ ⎡ a1n b11 ⎢ .. ⎥ ⎥ + ⎢ .. . ⎦ ⎣ . bm1 amn

⎡

a11 . . . ⎢ .. . . λ⎣ . . am1 . . .

... .. . ...

⎤ ⎡ b1n a11 + b11 ⎢ .. ⎥ .. ⎥=⎢ . ⎦ ⎣ . am1 + bm1 bmn

⎤ . . . a1n + b1n ⎥ .. .. ⎥, . ⎦ . . . . amn + bmn

⎤ ⎤ ⎡ a1n λa11 . . . λa1n . ⎥ .. ⎥ = ⎢ .. . . . .. ⎦ . . ⎦ ⎣ . amn λam1 . . . λamn

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A Some Properties of Vectors and Matrices

Example 2 Consider the matrices A=



   2 −1 3 0 −1 3 ,B = . −2 5 0 4 4 −2

Then we have A+B =



     2 −2 6 10 −5 15 0 1 −3 , 5A = , −B = . 2 9 −2 −10 25 0 −4 −4 2

The matrix operations satisfy the following properties: A+B = B +A A + (B + C) = (A + B) + C λ(A + B) = λA + λB. If A = (aij ) is an m × n matrix, and B = (bj k ) is an n × p matrix, then the product AB is defined as the m × p matrix given by C = (cik ) where cik =

n

aij bj k .

j =1

That is, the (i, k)-element of C is the scalar product of the ith row of A and the kth column of B. Note that the product of two matrices A and B is defined only when the number of columns of A is equal to the number of rows of B. Example 3 Let 

 2 −1 3 A= , −2 5 0

⎡

⎤ 2 −1 4 B = ⎣ 0 1 −1⎦ , −2 5 0

then AB =

  −2 12 9 . −4 7 −13

We can check that the matrix product satisfies the following properties: A(BC) = (AB)C A(B + C) = AB + AC A(λB) = λAB.

A

Some Properties of Vectors and Matrices

543

It is important to note that the matrix product is not commutative, that is, in general AB = BA. Moreover, if the product AB is defined, in general BA is not defined. A matrix is called a square matrix if the number of its rows is equal to the number of its columns. The main diagonal of an n×n square matrix A = (aij ) is the n-tuple (a11 , a22 , . . . , ann ). A square matrix D is called a diagonal matrix if all its elements are zero with the exception of its diagonal: ⎡ λ1 ⎢0 ⎢ D=⎢. ⎣ ..

0 λ2 .. .

... ... .. .

⎤ 0 0⎥ ⎥ .. ⎥ . .⎦

0 0 . . . λn

We also write D as D = diag(λ1 , λ2 . . . , λn ). Example 4 Let ⎡

⎤ 2 1 4 A = ⎣ 0 1 −1⎦ , −2 5 0

⎡ ⎤ 2 0 0 D = ⎣0 −1 0⎦ = diag(2, −1, 5). 0 0 5

Then ⎡ ⎤ ⎤ 4 −2 8 4 1 20 AD = ⎣ 0 −1 −5⎦ , DA = ⎣ 0 −1 1⎦ . −10 25 0 −4 −5 0 ⎡

The example above suggests the following fact, which is indeed true: If A = (v1 v2 . . . vn ) is a square matrix whose columns are the n-vectors v1 , v2 , . . . , vn , ⎡ ⎤ u1 ⎢u2 ⎥ ⎢ ⎥ and B = ⎢ . ⎥ is a square matrix whose rows are the n-vectors u1 , u2 , . . . , un , ⎣ .. ⎦ un

then

A diag(λ1 , λ2 , . . . , λn ) = (λ1 v1 λ2 v2 . . . λn vn ) ⎡ ⎤ λ1 u1 ⎢ λ2 u2 ⎥ ⎢ ⎥ diag(λ1 , λ2 , . . . , λn )B = ⎢ . ⎥ . ⎣ .. ⎦ λn un

(A.3a)

(A.3b)

544

A Some Properties of Vectors and Matrices

The n × n identity matrix is the matrix I diag(1, 1, . . . , 1), and satisfies AI = I A = A

(A.4)

for all n × n matrices A.

A.3 Systems of Linear Equations A system of m linear equations in n variables is a system of equations of the form a11 x1 + a12 x2 + · · · + a1n xn = b1 a21 x1 + a22 x2 + · · · + a2n xn = b2 ............... an1 x1 + an2 x2 + · · · + ann xn = bn We can write this system in the form Ax = b, where A = (aij ),

⎡ ⎤ x1 ⎢ ⎥ x = ⎣ ... ⎦ , xn

⎤ b1 ⎢ ⎥ b = ⎣ ... ⎦ . ⎡

bm

The information contained in the above system is also contained in the augmented matrix A|b with n + 1 rows and m columns. The solutions of the system are not changed by adding a multiple of one equation to another equation, multiplication of an equation by a non-zero constant, or interchanging two equations. This statement is equivalent to the statement that the set of solutions is not changed if the augmented matrix is subjected to a sequence of elementary row operations. There are three types of elementary row operations, namely 1. addition of a multiple of one row to another row, 2. multiplication of a row by a non-zero constant, 3. interchange of two rows. The system can be solved by reducing the matrix A to a triangular matrix using elementary row operations, a method called the Gauss–Jordan method. We demonstrate the method by an example. Example 1 Solve the system x − 3y − 5z = −8 −x + 2y + 4z = 5 2x − 5y − 11z = −9.

A

Some Properties of Vectors and Matrices

545

We add the first row to the second row, subtract double the first row from the third row, and multiply the new second row by −1. Next we add 3 times the second row to the first row and subtract the second row from the third row. Finally we subtract the third row from the first row, multiply the third row by −1/2, and subtract the third row from the second row. These operations give the sequence of augmented matrices ⎤ ⎡ ⎤⎞ ⎛⎡ ⎤ ⎡ ⎤⎞ ⎛⎡ 1 −3 −5  −8 1 −3 −5  −8 ⎝⎣−1 2 4 ⎦ ⎣ 5 ⎦⎠ ∼ ⎝⎣0 1 1 ⎦ ⎣ 3 ⎦⎠   0 1 −1  7 2 −5 −11  −9 ⎛⎡ ⎤  ⎡ ⎤⎞ 1 0 −2  1 ∼ ⎝⎣0 1 1 ⎦ ⎣3⎦⎠ 0 0 −2  4 ⎛⎡ ⎤  ⎡ ⎤⎞ 1 0 0  −3 ∼ ⎝⎣0 1 0⎦ ⎣ 5 ⎦⎠ . 0 0 1  −2 Then we may read off the solution from the final form as x = −3, y = 5, z = −2.

A.4 The Inverse Matrix Let A be a square n × n matrix. The inverse of A, if it exists, is a matrix A−1 such that AA−1 = A−1 A = I. Not every matrix A has an inverse. If A has an inverse, then it is called invertible, and its inverse is unique. Example 1 Find the inverse of the matrix ⎡

⎤ 1 −3 −5 A = ⎣−1 2 4 ⎦ . 2 −5 −11 Solution The inverse of A must be of the form (A.2), so we have to solve the system of 9 linear equations A · (xij ) = I.

546

A Some Properties of Vectors and Matrices

As in the previous example, we use the Gauss–Jordan method to solve this system of equations. The first and last steps of the algorithm are ⎤⎞ ⎛⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎞ 1 0 0  −1 −4 −1 1 −3 −5  1 0 0 ⎝⎣−1 2 4 ⎦ ⎣0 1 0⎦⎠ ∼ ⎝⎣0 1 0⎦ ⎣− 3 − 1 1 ⎦⎠ .   2 2 2 0 0 1  12 − 21 − 12 2 −5 −11  0 0 1 ⎛⎡

Therefore the inverse of A is the matrix A−1

⎤ −1 −4 −1 = ⎣− 23 − 12 12 ⎦ . 1 1 1 2 −2 −2 ⎡

A.5 Determinants The determinant of a square matrix is defined inductively by   a11 a12 = a11 a22 − a12 a21 det a21 a22 ⎡

a11 ⎢a21 ⎢ det ⎢ . ⎣ .. an1

a12 a22 .. . an2

... ... .. . ...

⎤ ⎡ ⎤ a1n a22 . . . a2n ⎥ a2n ⎥ ⎢ . . . ⎥ ⎥ .. ⎥ = a11 det ⎢ ⎣ .. . . .. ⎦ ⎦ . an2 . . . ann ann ⎤ ⎡ a21 . . . a2n ⎢ . . . ⎥ ⎥ −a12 det ⎢ ⎣ .. . . .. ⎦ + . . . an1 . . . ann ⎤ ⎡ a21 a22 . . . a2(n−1) ⎢ . . . .. ⎥ ⎥ +(−1)n+1 a1n det ⎢ ⎣ .. .. . . . ⎦. an1 an2 . . . an(n−1)

Example 1 Calculate the determinant of the matrix ⎡ ⎤ 2 0 1 A = ⎣1 4 1 ⎦ . 0 −2 −1

A

Some Properties of Vectors and Matrices

547

Solution We have 

   4 1 1 4 detA = 2 det − 0 + det = 2(−4 + 2) + (−2) = −6. −2 −1 0 −2 Theorem A.1 det(v1 . . . αu + βv . . . vn ) = α det(v1 . . . u . . . vn ) + β det(v1 . . . v . . . vn ) det(v1 . . . vi . . . vj . . . vn ) = − det(v1 . . . vj . . . vi . . . vn ) From (A.5), we can deduce the following properties of the determinant. det(v1 . . . 0 . . . vn ) = 0 det(v1 . . . u . . . u . . . vn ) = 0

(A.5)

det(v1 . . . u . . . v + αu . . . vn ) = det(v1 . . . u . . . v . . . vn ). The transpose of a matrix is the matrix whose columns are the rows of A, and whose rows are the columns of A. If A is an m × n matrix, then AT is an n × m matrix. One can show inductively that det AT = det A.

(A.6)

From (A.6) we conclude that the alternating multilinear properties (A.5-A.5) also hold for the rows of a matrix. Example 2 Calculate the determinant of the matrix ⎡

1 0 4 ⎢0 4 2 ⎢ ⎢ A = ⎢−2 1 −1 ⎢ ⎣ 10 4 −2 4 6 −1

3 −2 3 0 0

⎤ −1 0⎥ ⎥ ⎥ 2 ⎥. ⎥ 1⎦ 3

Solution We use properties (A.5–A.5) applied to the rows of A to calculate this determinant. ⎡ ⎤ ⎡ ⎤ 10 4 3 −1 10 4 3 −1 ⎢0 1 7 ⎢ 0 4 2 −2 0 ⎥ 9 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ det A = det ⎢ 0 1 7 9 0 ⎥ = − det ⎢ 0 4 2 −2 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0 4 −42 −30 11 ⎦ ⎣ 0 4 −42 −30 11 ⎦ 0 6 −17 −12 7 0 6 −17 −12 7

548

A Some Properties of Vectors and Matrices

⎡

⎤ 10 4 3 −1 ⎡ ⎤ ⎢0 1 7 −26 −38 0 9 0 ⎥ ⎢ ⎥ ⎢ ⎥ = − det ⎢ 0 0 −26 −38 0 ⎥ = − det ⎣ −70 −66 11 ⎦ ⎢ ⎥ ⎣ 0 0 −70 −66 11 ⎦ −59 −66 7 0 0 −59 −66 7 ⎡ ⎤       13 19 0 −66 11 −70 11 ⎣ ⎦ = 2 det −70 −66 11 = 2 13 det − −19 det −66 7 −59 7 −59 −66 7

= 2(13(−462 + 726) − 19(−490 + 649)) = 822.

Theorem A.2 If A and B are square matrices of the same size, then det(AB) = (det A)(det B).

(A.6a)

If the matrix A has an inverse A−1 , then Eq. (A.6a) implies (det A)(det A−1 ) = 1.

(A.7)

Thus, from Eq. (A.7), we can conclude that if A has an inverse, then det A = 0. The converse is true and is contained in the following theorem. Theorem A.3 Let A be an n × n matrix. Then the following are equivalent: 1. The system Ax = b has a unique solution for each n-vector b. 2. The matrix A is invertible. 3. det A = 0. Note that if det A = 0, then the system Ax = 0 has non-zero solutions x. We use this fact in the following section.

A.6 Eigenvalues and Eigenvectors Let A be a square matrix. We say that v = 0 is an eigenvector of A if λ is a scalar such that Av = λv

(A.8)

for some possibly complex scalar λ. The scalar λ is called an eigenvalue of A with respect to v. Note that each eigenvector of a matrix A corresponds to a unique eigenvalue; however, several eigenvectors may correspond to a single eigenvalue.

A

Some Properties of Vectors and Matrices

549

Example 1 Consider ⎡

⎤ 5 8 −3 A = ⎣0 −3 0 ⎦ , 0 0 2

⎡ ⎤ 1 v = ⎣0 ⎦ . 1

Then v is an eigenvector of A with respect to the eigenvalue λ = 2: ⎡ ⎤ ⎤⎡ ⎤ ⎡ ⎤ ⎡ 1 2 1 5 8 −3 Av = ⎣0 −3 0 ⎦ ⎣0⎦ = ⎣0⎦ = 2 ⎣0⎦ . 1 2 1 0 0 2 We now describe an algorithm to find the eigenvalues of a matrix. First, note that if v is an eigenvector of A with respect to the eigenvalue λ, then λ and v are solutions to Eq. (A.8) with v = 0. Equation (A.8) can be written in the form (A − λI )v = 0,

(A.9)

where I is the identity matrix of the same size as A. By Theorem A.3, Eq. (A.9) has a non-zero solution in v if and only if det(A − λI ) = 0.

(A.10)

Equation (A.10) is called the characteristic equation of the matrix A. Observe that if A is an n×n matrix, then the characteristic equation (A.10) is a polynomial equation in λ of degree n. Example 2 Calculate the eigenvalues and eigenvectors of the matrix ⎡ ⎤ 5 8 −3 A = ⎣0 −3 0 ⎦ . 0 0 2 Solution The characteristic equation of A is given by ⎡

⎤ 5−λ 8 −3 0 = det(A − λI ) = det ⎣ 0 −3 − λ 0 ⎦ = (5 − λ)(−3 − λ)(2 − λ). 0 0 2−λ

550

A Some Properties of Vectors and Matrices

Thus the eigenvalues of A are λ1 = 5, λ2 = −3, and λ3 = 2. To calculate the eigenvectors, we solve the Eq. (A.9) for each λi . Using the Gauss–Jordan method we obtain that ⎡ ⎤ 1 ⎣ v1 = 0⎦ , 0

⎡

⎤ 1 v2 = ⎣−1⎦ , 0

⎡ ⎤ 1 ⎣ v3 = 0 ⎦ 1

are the eigenvectors of A with respect to λ1 = 5, λ2 = −3, and λ3 = 2. Example 3 Calculate the eigenvalues and eigenvectors of the matrix ⎡

⎤ 110 A = ⎣2 0 0 ⎦ . 003 Solution The characteristic equation of A is given by ⎡ ⎤     1−λ 1 0 −λ 0 2 0 ⎣ ⎦ = (1 − λ) det 0 = det − det 2 −λ 0 0 3−λ 0 3−λ 0 0 3−λ = (1 − λ)(−λ)(3 − λ) − 2(3 − λ) = −(λ − 3)(λ + 1)(λ − 2).

Thus, the eigenvalues of A are then λ1 = 3, λ2 = −1, and λ3 = 2. The eigenvectors of A with respect to these eigenvalues are, respectively, ⎡ ⎤ 0 v1 = ⎣0⎦ , 1

⎡

⎤ −1 v2 = ⎣ 2 ⎦ , 0

⎡ ⎤ 1 v3 = ⎣1⎦ . 0

Example 4 Find the eigenvalues and eigenvectors of the matrix 

 1 3 A= . −4 2 Solution The characteristic equation of A is the equation 0 = det

  1−λ 3 = λ2 − 3λ + 14. −4 2 − λ

Then the eigenvalues of A are λ1 =

3 + 2

√

47 3 i and λ2 = − 2 2

√ 47 i. 2

A

Some Properties of Vectors and Matrices

551

The eigenvectors of A corresponding to λ1 and λ2 are, respectively, v1 =



1 2

+

3√

47 2 i



,

v1 =



1 2

−

3√

47 2 i



.

Let A be an n × n matrix, and suppose that the eigenvalues λ1 , λ2 , . . . , λn are all different. If v1 , v2 , . . . , vn are the eigenvectors of A with respect to the λi , let D = diag(λ1 , λ2 , . . . , λn ), S = (v1 v2 . . . vn ). The matrix D is the diagonal matrix whose diagonal elements are the eigenvalues of A, and S is the matrix whose columns are the eigenvectors of A. Hence AS = A(v1 v2 . . . vn ) = (Av1 Av2 . . . Avn ) = (λ1 v1 λ2 v2 . . . λn vn ) = (v1 v2 . . . vn ) diag(λ1 , λ2 , . . . , λn ) = SD, where we have used (A.3a). One can show that the matrix S has an inverse. Then, we factor A in the form A = SDS −1 .

(A.11)

The expression (A.11) is called the diagonalization of A. Not every matrix has a diagonalization. If the matrix A has a diagonalization, then we say that A is diagonalizable. We have seen, in the case where all the eigenvalues of A are distinct, that A is diagonalizable. Example 5 Diagonalize the matrix ⎡

⎤ 110 A = ⎣2 0 0 ⎦ . 003 Solution The eigenvalues of A are λ1 = 3, λ2 = 2, and λ3 = −1, and the eigenvectors with respect to these eigenvalues are ⎡ ⎤ 0 v1 = ⎣0⎦ , 1

⎡ ⎤ 1 v2 = ⎣1⎦ , 0

⎡ ⎤ −1 v3 = ⎣ 2 ⎦ . 0

552

A Some Properties of Vectors and Matrices

Let ⎤ ⎡ ⎡ ⎤ 30 0 01 1 S = ⎣0 1 −2⎦ , D = ⎣0 2 0 ⎦ . 0 0 −1 10 0 Then we have A = SDS −1 . Thus, the diagonalization of A is ⎤⎡ ⎤⎡ ⎤ ⎡ 01 1 30 0 0 0 110 ⎣2 0 0⎦ = ⎣0 1 −2⎦ ⎣0 2 0 ⎦ ⎣2/3 1/3 10 0 0 0 −1 1/3 −1/3 003 ⎡

⎤ 1 0⎦ . 0

From (A.3) one can check that

diag(λ1 , λ2 , . . . , λn )

Thus, if A = SDS −1 ,

k

= diag(λk1 , λk2 , . . . , λkn ).

Ak = (SDS −1 )(SDS −1 ) · · · (SDS −1 )

(k factors)

= SD k S −1 .

If p(x) = ak x k + . . . + a1 x + a0 is a polynomial and A is an n × n matrix, we define p(A) = ak Ak + . . . + a1 A + a0 I. By the above observations, we see that if A is diagonalizable, then p(A) = p(SDS −1 ) = S · p(D) · S −1 = S · diag(p(λ1 ), p(λ2 ), . . . , p(λn )) · S −1 . Example 6 Calculate p(A) where p(x) = x 2 − 3x + 14, and A=



 1 3 . −4 2

Solution The characteristic equation of A is λ2 − 3λ + 14 = 0. Therefore p(A) = S · p(D) · S −1 = S · diag(p(λ1 ), p(λ2 )) · S −1 = S · diag(0, 0) · S −1 = 0. The above example suggests the following fact, which is indeed true: Every diagonalizable matrix satisfies its characteristic equation. In fact, one can show that every matrix satisfies its characteristic equation.

Appendix B

First Order Ordinary Differential Equations

B.1 Exponential Growth and Decay The rate of change of some quantity is often proportional to the amount of the quantity present. This may be true, for example, of the size of a population with enough resources that its growth is unrestricted, and depends only on an inherent per capita reproductive rate. It can also apply to a decaying population—for example, the mass of a piece of a radioactive substance. In such a case, if y(t) is the quantity at time t, then y(t) satisfies the differential equation dy = ay dt

(B.1)

where a is a constant representing the proportional growth or decay rate with a positive if the quantity is increasing and negative if the quantity is decreasing. It is easy to verify that y = ceat is a solution of the differential equation (B.1) for every choice of the constant c. By this we mean that if we substitute the function y = ceat into the differential equation (B.1) it becomes an identity. If y = ceat , then y ′ = aceat = ay, and this is the necessary verification. Thus the differential equation (B.1) has an infinite family of solutions (one for every choice of the constant c, including c = 0), namely y = ceat .

(B.2)

In order for a mathematical problem to be a plausible description of a scientific situation, the mathematical problem must have only one solution; if there were multiple solutions, we would not know which solution represents the situation. This suggests that the differential equation (B.1) by itself is not enough to specify a description of a physical situation. We must also specify the value of the function y for some initial time when we may measure the quantity y and then allow the © Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9

553

554

B

First Order Ordinary Differential Equations

system to start running. For example, we might impose the additional requirement, called an initial condition, that y(0) = y0 .

(B.3)

A problem consisting of a differential equation together with an initial condition is called an initial value problem. We may determine the value of c for which the solution (B.2) of the differential equation (B.1) also satisfies the initial condition (B.3) by substituting t = 0, y = y0 into the form (B.2). This gives the equation y0 = c e0 = c and thus c = y0 . We now use this value of c to give the solution of the differential equation (B.1) which also satisfies the initial condition (B.3), namely y = y0 eat . In order to show that this is the only solution of the initial value problem (B.1), (B.3), we must show that every solution of the differential equation (B.1) is of the form (B.2). To prove this, suppose that y(t) is a solution of the differential equation (B.1), that is, that y ′ (t) = ay(t) for every value of t. If y(t) = 0, division of this equation by y(t) gives y ′ (t) d = log |y(t)| = a. y(t) dt

(B.4)

Integration of both sides of (B.4) gives log |y(t)| = at + k for some constant of integration k. Then |y(t)| = eat+k = ek eat . Because eat and ek are positive for every value of t, |y(t)| cannot be zero, and thus y(t) cannot change sign. We may remove the absolute value and conclude that y(t) is a constant multiple of eat , y = ceat . We note also that if y(t) is different from zero for one value of t then y(t) is different from zero for every value of t. Thus the division by y(t) at the beginning of the proof is legitimate unless the solution y(t) is identically zero. The identically zero function is a solution of the differential equation, as is easily verified by substitution, and it is contained in the family of solutions y = ceat with c = 0. The absolute value which appears in the integration produces some complications which may be avoided if we know that the solution must be non-negative, so that |y(t)| = y(t), as is the case in many applications. If we know that a solution y(t) of the differential equation y ′ = ay is positive for all t, we could replace (B.4) by d log y(t) = a dt and then integrate to obtain log y(t) = at + k, y(t) = eat+k = eat ek = ceat .

B

First Order Ordinary Differential Equations

555

The logical argument in the above proof is that if the differential equation has a solution, then that solution must have a certain form. However, it also derives the form and thus serves as a method of determining the solution. We now have a family of solutions of the differential equation (B.1). In order to determine the member of this family which satisfies a given initial condition, that is, in order to determine the value of the constant c, we merely substitute the initial condition into the family of solutions. This procedure may be followed in any situation which is described by an initial value problem, including population growth models and radioactive decay. Example 1 Suppose that a given population of protozoa develops according to a simple growth law with a growth rate of 0.6 per member per day, that there are no deaths, and that on day zero the population consists of two members. Find the population size after 10 days. Solution The population size satisfies the differential equation (B.1) with a = 0.6, and is therefore given by y(t) = ce0.6t . Since y(0) = 2, we substitute t = 0, y = 2, and we obtain 2 = c. Thus the solution which satisfies the initial condition is y(t) = 2 e0.6t , and the population size after 10 days is y(10) = 2 e(0.6)(10) = 403 (with population size rounded off to the nearest integer).  If we know that a population grows exponentially according to an exponential growth law but do not know the rate of growth we view the solution y = ceat as containing two parameters which must be determined. This requires knowledge of the population size at two different times to provide two equations which may be solved for these two parameters. Example 2 Suppose that a population which follows an exponential growth law has 50 members at a starting time and 100 members at the end of 10 days. Find the population at the end of 20 days. Solution The population size at time t satisfies y(t) = ceat and y(0) = 50, y(10) = 100. Thus y(0) = 50 = ce0 , y(10) = ce10a = 100. It follows that 2 c = 50 and 100 = 50e10a . We obtain e10a = 2, a = log 10 = 0.0693. Finally, we obtain y(20) = 50 e(20)(log 2)/10 = 50 e2 log 2 = 50 · 22 = 200.



B.2 Radioactive Decay Radioactive materials decay because a fraction of their atoms decompose into other substances. If y(t) represents the mass of a sample of a radioactive substance at time t, and a fraction k of its atoms decompose in unit time, then y(t + h) − y(t) is approximately −ky(t) and we are led to the differential equation (B.1) with a replaced by −k. If it is clear from the nature of the problem that the constant of

556

B

First Order Ordinary Differential Equations

proportionality must be negative, we will use −k for the constant of proportionality, giving a differential equation y ′ = −ky

(B.5)

with k > 0. Example 3 The radioactive element strontium 90 has a decay constant 2.48 × 10−2 years−1 . How long will it take for a quantity of strontium 90 to decrease to half of its original mass? Solution The mass y(t) of strontium 90 at time t satisfies the differential equation (B.7) with k = 2.48 × 10−2 . If we denote the mass at time t = 0 by y0 , then −2 y(t) = y0 e−(2.48×10 )t . The value of t for which y(t) = y0 /2 is the solution of y0 −2 = y0 e−(2.48×10 )t . 2 If we divide both sides of this equation by y0 and then take natural logarithms, we have −(2.48 × 10−2 )t = log

1 = − log 2 2

so that t = (log n)/(2.48 × 10−2 ) = 27.9 years.



The time required for the mass of a radioactive substance to decrease to half of its starting value is called the half-life of the substance. The half-life T is related to the decay constant k by the equation T =

log 2 k

because if y(t) = y0 e−kt and (by definition) y(T ) = y20 , then e−kT = 12 , so that −kT = log 21 = − log 2. For radioactive substances it is common to give the halflife rather than the decay constant. Example 4 Radium 226 is known to have a half-life of 1620 years. Find the length of time required for a sample of radium 226 to be reduced to one fourth of its original size. log 2 Solution The decay constant for radium 226 is k = 1620 = 4.28 × 10−4 years−1 . −kt In terms of k, the mass of a sample at time t is y0 e if the starting mass is y0 . The time τ at which the mass is y0 /4 is obtained by solving the equation y0 /4 = y0 e−kτ or e−kτ = 1/4. Taking natural logarithms we obtain −kτ = log 4, which gives

τ =−

log 34 1620(log 34 ) = = 672 years. k log 2



B

First Order Ordinary Differential Equations

557

The radioactive element carbon 14 decays to ordinary carbon (carbon 12) with a decay constant 1.244 × 10−4 years−1 , and thus the half-life of carbon 14 is 5570 years. This has an important application, called carbon dating, for determining the approximate age of fossil materials. The carbon in living matter contains a small proportion of carbon 14 absorbed from the atmosphere. When a plant or animal dies, it no longer absorbs carbon 14 and the proportion of carbon 14 decreases because of radioactive decay. By comparing the proportion of carbon 14 in a fossil with the proportion assumed to have been present before death, it is possible to calculate the time since absorption of carbon 14 ceased. Example 5 Living tissue contains approximately 6 × 1010 atoms of carbon 14 per gram of carbon. A wooden beam in an ancient Egyptian tomb from the First Dynasty contained approximately 3.33 × 1010 atoms of carbon 14 per gram of carbon. How old is the tomb? Solution The number of atoms of carbon 14 per gram of carbon, y(t), is given by y(t) = y0 e−kt , with y0 = 6 × 1010 , k = 1.244 × 10−4 , and y(t) = 3.33 × 1010 for this particular t value. Thus the age of the tomb is given by the solution of the equation e−(1.244×10

−4 )t

=

3.33 3.33 × 1010 , = 6 6 × 1010

and if we take natural logarithms this reduces to t =−

log 3.33 − log 6 = 4733 years. 1.244 × 10−4



B.3 Solutions and Direction Fields By a differential equation we will mean simply a relation between an unknown function and its derivatives. We will confine ourselves to ordinary differential equations, which are differential equations whose unknown function is a function of one variable so that its derivatives are ordinary derivatives. A partial differential equation is a differential equation whose unknown function is a function of more than one variable, so that the derivatives involved are partial derivatives. The order of a differential equation is the order of the highest-order derivative appearing in the differential equation. In this chapter we shall consider first-order differential equations, relations involving an unknown function y(t) and its first derivative y ′ (t) = dy dt . The general form of a first-order differential equation is y′ =

dy = f (t, y), dt

with f a given function of the two variables t and y.

(B.6)

558

B

First Order Ordinary Differential Equations

By a solution of the differential equation (B.6) we mean a differentiable function y of t on some t-interval I such that, for every t in the interval I , y ′ (t) = f (t, y(t)). In other words, differentiating the function y(t) results in the function f (t, y(t)). For example, we have seen in the preceding section that, whatever the value of the constant c, the function y = ceat is a solution of the differential equation y ′ = ay on every t-interval. We see this by differentiating ceat to get a ceat , which we can rewrite as ay. To verify whether a given function is a solution of a given differential equation, we need only substitute into the differential equation and check whether it then reduces to an identity. Example 1 Show that the function y = y ′ = −y 2 .

1 t+1

is a solution of the differential equation

Solution For the given function, dy 1 =− = −y 2 dt (t + 1)2 and this shows that it is indeed a solution.



In the same way we can verify that a family of functions satisfies a given differential equation. By a family of functions we will mean a function which includes an arbitrary constant, so that each value of the constant defines a distinct 5t 5t 5t 5t function. √ 5t The family ce , for instance, includes the functions e , −4e , 12e , and 3 e , among others. When we say that a family of functions satisfies a differential equation, we mean that substitution of the family (i.e., the general form) into the differential equation gives an identity satisfied for every choice of the constant. Example 2 Show that for every c the function y = differential equation y ′ = −y 2 .

1 t+c

is a solution of the

Solution For the given function, dy 1 =− = −y 2 , dt (t + c)2 and thus each member of the given family of functions is a solution.



In applications we are usually interested in finding not a family of solutions of a differential equation but a solution which satisfies some additional requirement. In the various examples in Sect. B.1 the additional requirement was that the solution should have a specified value for a specified value of the independent variable t. Such a requirement, of the form

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559

y(t0 ) = y0

(B.7)

is called an initial condition, and t0 is called the initial time while y0 is called the initial value. A problem consisting of a differential equation (B.1) together with an initial condition (B.2) is called an initial value problem. Geometrically, an initial condition picks out the solution from a family of solutions which passes through the point (t0 , y0 ) in the t-y plane. Physically, this corresponds to measuring the state of a system at the time t0 and using the solution of the initial value problem to predict the future behavior of the system. Example 4 Find the solution of the differential equation y ′ = −y 2 of the form 1 which satisfies the initial condition y(0) = 1. y = t+c 1 Solution We substitute the values t = 0, y = 1 into the equation y = t+c , and 1 we obtain a condition on c, namely 1 = c , whose solution is c = 1. The required 1 .  solution is the function in the given family with c = 1, namely y = t+1

Example 5 Find the solution of the differential equation y ′ = −y 2 which satisfies the general initial condition y(0) = y0 , where y0 is arbitrary.

1 Solution We substitute the values t = 0, y = y0 into the equation y = t+c and 1 1 solve the resulting equation y0 = c for c, obtaining c = y0 provided y0 = 0. Thus the solution of the initial value problem is

y=

1 (t +

1 2 y0 )

except if y0 = 0. If y0 = 0, there is no solution of the initial value problem of the given form; in this case the identically zero function, y = 0, is a solution. We have now obtained a solution of the initial value problem with arbitrary initial value at y = 0 for the differential equation y ′ = −y 2 .  A family of solutions may arise if we are considering a differential equation with no initial condition imposed, and we will then also be concerned with the question of whether the given family contains all solutions of the differential equation. To answer this question, we will need to make use of a theorem which guarantees that each initial value problem for the given differential equation has exactly one solution. More specifically, if an initial value problem is to be a usable mathematical description of a scientific problem, it must have a solution, for otherwise it would be of no use in predicting behavior. Furthermore, it should have only one solution, for otherwise we would not know which solution describes the system. Thus for applications it is vital that there be a mathematical theory telling us that an initial value problem has exactly one solution. Fortunately, there is a very general theorem which tells us that this is true for the initial value problem (B.6), (B.3) provided

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the function f is reasonably smooth. We will state this result and ask the reader to accept it without proof because the proof requires more advanced mathematical knowledge than we have at present. Existence and Uniqueness Theorem If the function f (t, y) is differentiable with respect to y in some region of the plane which contains the point (t0 , y0 ), then the initial value problem consisting of the differential equation y ′ = f (t, y) and the initial condition y(t0 ) = y0 has a unique solution which is defined on some tinterval containing t0 in its interior. Even though the function f (t, y) may be well-behaved in the whole t-y plane, there is no assurance that a solution will be defined for all t. As we have seen 1 of y ′ = −y 2 , y(0) = 1 exists only for in Example 1, the solution y = t+1 −1 < t < ∞. As we have seen in Example 2, each solution of the family of 1 has a different interval of existence. In Example 5, we have solutions y = t+c shown how to rewrite a family of solutions for a differential equation in terms of an arbitrary initial condition—that is, as a solution of an initial value problem for that differential equation. We have also seen how to identify those initial conditions which cannot be satisfied by a member of the given family. Often there are constant functions which are not members of the given family but which are solutions and satisfy initial conditions that cannot be satisfied by a member of the family. The existence and uniqueness theorem tells us that if we can find a family of solutions, possibly supplemented by some additional solutions, so that we can find this collection contains a solution corresponding to each possible initial condition, then we have found the set of all solutions of the differential equation. A differential equation which arises in various applications, including models for population growth and spread of rumors, is the logistic differential equation,  y y ′ = ry 1 − K

(B.8)

containing two parameters r and K. The basic idea behind this form is that instead of a constant per capita growth rate r as in the exponential growth equation it is more realistic to assume that the per capita growth rate decreases as the population size increases. The form 1 − Ky used in the logistic equation is the simplest form for a decreasing per capita growth rate. We may verify that for every constant c the function y=

K 1 + ce−rt

(B.9)

is a solution of this differential equation. To see this, note that for the given function y, y′ =

Kcr e−rt (1 + ce−rt )2

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561

and 1−

K −y ce−rt y = = . K K (1 + ce−rt )

Thus  y Krc e−rt ry 1 − = = y′, K (1 + ce−rt )2 and the given function satisfies the logistic equation for every choice of c. To find the solution which obeys the initial condition y(0) = y0 , we substitute K 0 = y0 which implies c = K−y t = 0, y = y0 into the form (B.9), obtaining 1+c y0 as long as y0 = 0 and gives the solution y=

1+



K Ky0  = K − y0 −rt y0 + (K − y0 ) e−rt e y0

(B.10)

to the initial value problem with y0 = 0. Note that the denominator begins at K (for t = 0) and moves toward y0 as t → ∞. Now suppose that K represents some physical quantity such that K > 0. It is easy to see from the form (B.10) that if y0 > 0, then the solution y(t) exists for all t > 0, and limt→∞ y(t) = K. If y0 < 0, then this solution does not exist for all t > 0, because y(t) → −∞ where the −y0 denominator changes sign: as y0 + (K − y0 )e−rt → 0, or t → − log( K−y ). If 0 y0 = 0, the solution of the initial value problem is not given by (B.10), but is the identically zero function y = 0. We observe that the family of solutions (B.9) of the logistic differential equation (B.8) includes the constant solution y = K (with c = 0) but not the constant solution y = 0. The existence and uniqueness theorem shows that, since we have now obtained a solution corresponding to each possible initial condition, we have obtained all solutions of the logistic differential equation. The geometric interpretation of a solution y(t) to a differential equation (B.6) is that the curve y = y(t) has slope f (t, y) at each point (t, y) along its length. Thus we might think of approximating the solution curve by piecing together short line segments whose slope at each point (t, y) is f (t, y). To realize this idea, we construct at each point (t, y) in some region of the plane a short line segment with slope f (t, y). The collection of line segments is called the direction field of the differential equation (B.6). The direction field can help us to visualize solutions of the differential equation since at each point on its graph a solution curve is tangent to the line segment at that point. We may sketch the solutions of a differential equation by connecting these line segments by smooth curves. A direction field and some solutions of the differential equation y ′ = y are shown in Fig. B.1. The direction field suggests exponential solutions, which we know from Sect. B.1 to be correct.

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Fig. B.1 Direction field and solutions for y ′ = y

A direction field and some solutions of the differential equation y ′ = y(1−y) are shown in Fig. B.2. The direction field indicates that solutions below the t-axis are unbounded below while solutions above the t-axis tend to 1 as t → ∞, consistent with what we have established for this logistic differential equation. Drawing direction fields by hand is a difficult and time-consuming task. There are computer programs, both self-contained and portions of more elaborate computational systems such as Maple, Matlab, and Mathematica, which can generate direction fields for a differential equation and can also sketch solution curves corresponding to these direction fields. The examples here have been produced by Maple. The geometric view of differential equations presented by the direction field will appear again when we examine some qualitative properties in Sect. B.5.

B.4 Equations with Variables Separable In this section, we shall learn a method for finding solutions of a class of differential equations. The method is based on the method we used in Sect. B.1 for solving the differential equation of exponential growth or decay, and is applicable to differential equations with variables separable. A differential equation y ′ = f (t, y) is called separable or is said to have variables separable if the function f can be expressed

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563

Fig. B.2 Direction field and solutions for y ′ = y(1 − y)

in the form f (t, y) = p(t)q(y).

(B.11)

In particular, if f is independent of t and is a function of y only, the differential equation (B.6) is separable; such an equation is said to be autonomous. The reason for the name separable is that the differential equation can then be written as y′ = p(t) q(y) with all the dependence on y on the left and all the dependence on t on the righty hand side of the equation, provided that q(y) = 0. For example, y ′ = 1+t 2 and ′ 2 ′ y = y are both separable, whereas y = sin t − 2ty is not separable. The examples discussed in Sect. B.1 are also separable and the method of solution described for Eq. (B.1) of Sect. B.1 is a special case of the general method of solution to be developed for separable equations. Before explaining this general technique, let us work out some more examples.

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Example 1 Find all solutions of the differential equation y ′ = −y 2 . Solution We divide the equation by y 2 , permissible if y = 0, to give 1 dy = −1. y 2 dt We then integrate both sides of this equation with respect to t. In order to integrate the left side of the equation we must make the substitution y = y(t), where y(t) is the (as yet unknown) solution. This substitution gives 

1 dy dt = y 2 dt



dy 1 =− +c y y2

and we obtain −

1 = −t − c y

(B.12)

1 or y = t+c , with c a constant of integration. Observe that no matter what value of c is chosen this solution is never equal to zero. We began by dividing the equation by y 2 , which was legitimate provided y = 0. If y = 0, we cannot divide, but the constant function y = 0 is a solution. We now have all the solutions of the differential equation, namely the family (B.12) together with the zero solution. 

Example 2 Solve the initial value problem y ′ = −y 2 ,

y(0) = 1.

Solution We begin by using a more colloquial form of separation of variables than the procedure of Example 1. We write the differential equation in the form dy dt = −y 2 and separate variables by dividing the equation through by y 2 to give −

dy = dt. y2

This form of the equation is meaningless without an interpretation of differentials, but the integrated form   dy = dt − y2 is meaningful. Carrying out the integration, we obtain y1 = t + c, where c is a constant of integration. Since y = 1 for t = 0 we may substitute these values into the solution and obtain 1 = c. Substituting this value of c we now obtain

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First Order Ordinary Differential Equations

565

1 = t + 1. y 1 Solving for y, we obtain the solution y = t+1 , and we may easily verify that this is a solution of the initial value problem, defined for all t ≥ 0. 

The version of separation of variables used in Example 2 is easier to apply in practice than the more precise version used in Example 1. When we treat the general case we will justify this approach. Example 3 Solve the initial value problem y ′ = −y 2 ,

y(0) = 0.

1 Solution The procedure used in Example 1 leads to the family of solutions y = t+c for the differential equation y ′ = −y 2 . When we substitute t = 0, y = 0 and attempt to solve for the constant c, we find that there is no solution. When we divided the differential equation by y 2 we had to assume y = 0, but the constant function y ≡ 0 is also a solution of the differential equation, as may easily be verified. Since this function also satisfies the initial condition, it is the solution of the given initial value problem. 

Let us now apply the technique of Example 2 to show how to obtain the solution in the general case of a differential equation with variables separable. We will make use of the idea of the definition of the indefinite integral of a function as a new function whose derivative is the derivative of the given function. We first solve the differential equation y ′ = p(t)q(y)

(B.13)

where p is continuous on some interval a < t < b and q is continuous on some interval c < y < d. We solve the differential equation (B.13) by separating variables as in Example 1 dividing by q(y) and integrating to give 

dy = q(y)



p(t)dt + c

(B.14)

where c is a constant of integration. We now define Q(y) =



dy , q(y)

P (t) =



p(t)dt,

by which we mean that Q(y) is a function of y whose derivative with respect to 1 y is q(y) and P (t) is a function of t whose derivative with respect to t is p(t). Then (B.14) becomes

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First Order Ordinary Differential Equations

Q(y) = P (t) + c

(B.15)

and this equation defines a family of solutions of the differential equation (B.13). To verify that each function y(t) defined implicitly by (B.15) is indeed a solution of the differential equation (B.13), we differentiate (B.15) implicitly with respect to t, obtaining 1 dy = p(t) q(y(t)) dt on any interval on which q(y(t)) = 0, and this shows that y(t) satisfies the differential equation (B.13). There may be constant solutions of (B.13) which are not included in the family (B.15). A constant solution to the original equation (B.13) corresponds to a solution of the equation q(y) = 0. In order to find the one solution of the differential equation (B.13) which satisfies the initial condition y(t0 ) = y0 , it will help to be more specific in the choice of the 1 indefinite integrals Q(y) and P (t). We let Q(y) be the indefinite integral of q(y) such that Q(y0 ) = 0, and we let P (t) be the indefinite integral of p(t) such that P (t0 ) = 0. Then, because of the fundamental theorem of calculus, we have Q(y) =



y

y0

du , q(u)

P (t) =



t

p(s) ds

t0

and we may write the solution (B.15) in the form 

y y0

du = q(u)



t

p(s) ds.

t0

Now substitution of the initial conditions t = t0 , y = y0 gives c = 0. Thus the solution of the initial value problem is given implicitly by 

y

y0

du = q(u)



t

p(s) ds

(B.16)

t0

We have now solved the initial value problem in the case q(y0 ) = 0. Since y(t0 ) = y0 and the function q is continuous at y0 , q(y(t)) is continuous, and therefore q(y(t)) = 0 on some interval containing t0 (possibly smaller than the original interval a < t < b). On this interval, y(t) is the unique solution of the initial value problem. If q(y0 ) = 0, we have the constant function y = y0 as a solution of the initial value problem in place of the solution given by (B.16). We shall see in Sect. B.5 that the constant solutions of a separable differentiable equation (B.13) play an important role in describing the behavior of all solutions as t → ∞.

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First Order Ordinary Differential Equations

567

The differential equation y ′ = −ay + b

(B.17)

where a and b are given constants appears in some of the examples in Sect. B.4. In order to solve it, we separate variables, obtaining   dy = dt. −ay + b Integration gives 1 − log(−ay + b) = t + c a with c an arbitrary constant of integration. Algebraic solution now gives log(−ay + b) = −a(t + c)

−ay + b = e−a(t+c) = e−ac e−at ay = b − e−ac e−at y=

b e−ac −at − e . a a −ac

We now rename the arbitrary constant − e a as a new arbitrary constant, which we again call c, and obtain the family of solutions y=

b + ce−at . a

(B.18)

In our separation of variables, we overlooked the constant solution y = ab , but this solution is contained in the family (B.18). In order to find the solution of (B.17) which satisfies the initial condition y(0) = y0

(B.19)

we substitute t = 0, y = y0 into (B.18), obtaining y0 =

b +c a

or c = y0 − ab . This value of c gives the solution y=

b b b + (y0 − )e−at = (1 − e−at ) + y0 e−at a a a

of the initial value problem.

(B.20)

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The solutions (B.18) and (B.20) have been obtained without regard for the sign of the coefficients b and a. We will think of a in (B.18) and (B.20) as positive, and will rewrite the solutions for a negative by making the replacement r = −a, thinking of r as positive. The result of this is that the solutions of the differential equation y ′ = ry + b

(B.21)

are given by the family of functions b y = − + cert r

(B.22)

and the solution of the differential equation (B.21) which satisfies the initial condition (B.19) is b b y = − (1 − ert ) + y0 ert = (ert − 1) + y0 ert . r r

(B.23)

In applications, the specific form of the solution of a differential equation is often less important than the behavior of the solution for large values of t. Because e−at → 0 as t → ∞ when a > 0, we see from (B.18) that every solution of (B.17) tends to the limit ab as t → ∞ if a > 0. This is an example of qualitative information about the behavior of solutions which will be useful in applications. In Sect. B.5, we shall examine other qualitative questions—information about the behavior of solutions of a differential equation which may be obtained indirectly rather than by an explicit solution. To conclude this section, we return to the logistic differential equation (B.8) whose solutions were described in Sect. B.3. In Sect. B.3, we verified that the solutions were given by Eq. (B.9) but did not show how to obtain these solutions. The differential equation (B.8) is separable, and separation of variables and integration gives 

Kdy = y(K − y)



r dt

provided y = 0, y = K. In order to evaluate the integral on the left-hand side, we use the algebraic relation (which may be obtained by partial fractions) K 1 1 = + y(K − y) y K −y and then rewrite    K dy dy dy = + = log |y| − log |K − y| = rt + c, y(K − y) y K −y

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569

so   log 

y   = rt + c. K −y

We can now exponentiate both sides of the equation to obtain   

y   = ert+c = ert ec . K −y

If we remove the absolute value bars from the left-hand side, we can define a new constant C on the right-hand side, equal to e−c if 0 < y < K, and equal to −e−c otherwise, thus rewriting the right-hand side as C1 ert . Finally, we solve for y, and obtain the family of solutions y=

K 1 + Ce−rt

(B.24)

as in Sect. B.3. This family contains all solutions of (B.8) except for the constant solution y = 0. A qualitative observation is that if r > 0 every positive solution approaches the limit K as t → ∞. (The only other non-negative solution is the constant solution y ≡ 0). The explicit solution of the logistic differential equation is complicated, requiring some knowledge of techniques of integration (separation into partial fractions) as well as some algebraic manipulations to convert the implicit function given by integration into an explicit expression. In Sect. B.5, we will see how to obtain the qualitative observation that if r > 0, every positive solution approaches the limit K as t → ∞ without the need to go through these calculations.

B.5 Qualitative Properties of Differential Equations We have seen some instances in which all solutions of a differential equation, or at least all solutions with initial values in some interval, tend to the same limit as t → ∞. Two examples are the (separable) linear differential equation with constant coefficients y ′ = −ay + b

(B.25)

for which every solution approaches the limit ab as t → ∞ provided a > 0 (Sect. B.4), and the logistic differential equation (B.8) for which every solution with positive initial value approaches the limit K as t → ∞ (Sect. B.4) if r > 0. In applications we are often particularly interested in the long-term behavior of solutions, especially since many of the models we develop will be complex enough to make finding an explicit solution impractical. In this section, we describe some

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information of this nature which may be obtained indirectly, without explicit solution of a differential equation. Properties of solutions obtained without actually finding an expression for the solutions are called qualitative properties. We shall study only differential equations which do not depend explicitly on the independent variable t, of the general form y ′ = g(y).

(B.26)

Such differential equations are called autonomous. We will always assume that the function g(y) is sufficiently smooth that the existence and uniqueness theorem of Sect. B.3 is valid, and there is a unique solution of the differential equation (B.26) through each initial An autonomous differential equation is always separable,  point. dy is difficult or impossible to evaluate, solution by separation but if the integral g(y) of variables is impractical. In Sect. B.4, when we established the method of separation of variables, we pointed out the possibility of constant solutions which do not come from separation of variables. It will turn out that examination of these constant solutions is of central importance in the qualitative analysis. We begin with an example, the logistic equation already solved explicitly in Sect. B.3. Example 1 Determine the behavior as t → ∞ of solutions of the logistic equation (B.8) with y(0) > 0. Solution We write  y g(y) = ry 1 − , K and note that g(y) > 0 if 0 < y < K, g(y) < 0 if y > K. Since a solution y(t) of (B.8) has the property that y(t is increasing if g(y(t)) > 0, a solution with 0 < y(0) < K increases for all t such that y(t) < K. However, this solution must remain less than K for all t, because if there were a value t ∗ such that y(t ∗ ) = K, there would be two solutions with y(t ∗ ) = K, namely the solution y(t) and the constant solution y = K, and this would violate the uniqueness theorem. Thus y(t) is an increasing function for all t bounded above by K, and therefore tends to a limit y∞ as t → ∞. If y∞ < K, then lim g(y(t)) = g(y∞ ) > 0,

t→∞

and this is a contradiction because if y(t) is a monotone increasing differentiable function that tends to a limit, its derivative must approach zero. Therefore, we have shown that lim y(t) = K.

t→∞

A similar argument if y(0) > K shows that the solution y(t) decreases monotonically to the limit K, and we have shown that every solution of (B.8) with y(0) > 0

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571

Fig. B.3 Phase line for the logistic equation

approaches the limit K as t → ∞ using only information about the sign of the function g(y). We may describe this pictorially by drawing the phase line. This is the graph of the function g(y) together with arrows drawn along the y-axis to the right on intervals in which the function g(y) is positive (corresponding to intervals in which a solution y(t) is increasing), and arrows drawn to the left on intervals on which the function g(y) is negative. Points where g(y) = 0 correspond to constant solutions of the differential equation, and the arrows indicate which of these equilibrium points are approached by other solutions (the equilibrium y = K for the logistic equation), and which equilibrium points repel solutions beginning near them (the equilibrium y = 0 for the logistic equation (Fig. B.3).  The information given by the phase line is reflected in the graphs of solutions of the logistic equation (Fig. B.4). The analysis in general of the behavior as t → ∞ of the solutions of the differential equation (B.26) follows a similar pattern, determined from the nature of the constant solutions. The theory depends on the following properties of autonomous differential equations. For simplicity, we shall consider only nonnegative values of t, and we will think of solutions as determined by their initial values for t = 0. Property 1 If , y is a solution of the equation g(y) = 0, then the constant function y =, y is a solution of the differential equation y ′ = g(y). Conversely, if y = , y is a constant solution of the differential equation y ′ = g(y), then , y is a solution of the equation g(y) = 0.

To establish this property, we need only observe that because the derivative of a constant function is the zero function, a constant function y = , y is a solution y ) = 0. A solution , y of g(y) = 0 is called an equilibrium of (B.26) if and only if g(, or critical point of the differential equation (B.26), and the corresponding constant solution of (B.26) is called an equilibrium solution. The graphs of equilibrium solutions of an autonomous differential equation (B.26) are horizontal lines which separate the t − y plane into horizontal bands of the form {(t, y) | t ≥ 0, y1 < y < y2 }

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Fig. B.4 Some solutions of the logistic equation (B.8) with r = 2, K = 2

where y1 and y2 are consecutive equilibria of (B.26), with no equilibrium of (B.26) between y1 and y2 . Property 2 The graph of every solution curve of the differential equation y ′ = g(y) remains in the same band and is either monotone increasing [y ′ (t) > 0] or monotone decreasing [y ′ (t) < 0] for all t ≥ 0, depending on whether g(y) > 0 or g(y) < 0, respectively, in the band. Suppose that y1 and y2 are consecutive equilibria of (B.26) and that y1 < y(0) < y2 . If g(y) is indeed smooth enough to apply the existence and uniqueness theorem of Sect. B.3, then the graph of a solution cannot cross either of the constant solutions which form the boundaries of the band; otherwise at the crossing there would be a point in the t − y plane with two solutions passing through it, violating the uniqueness. Therefore the solution must remain in this band for all t ≥ 0. If, for example, g(y) > 0 for y1 < y < y2 , then y ′ (t) = g{y(t)} > 0 for t ≥ 0, and the solution y(t) is monotone increasing. A similar argument shows that if g(y) < 0 for y1 < y < y2 , the solution y(t) is monotone decreasing. If y(0) is above the largest equilibrium of (B.26), the band containing the solution is unbounded, and if g(y) > 0 in this band, then the solution y(t) may be (positively) unbounded and does not necessarily exist for all t ≥ 0. Likewise, if y(0) is below the

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573

smallest equilibrium of (B.26) and if g(y) < 0 in the band containing the solution, then the solution may be unbounded (negatively) and may fail to exist for all t ≥ 0. Property 3 Every solution of the differential equation y ′ = g(y) which remains bounded for 0 ≤ t < ∞ approaches a limit as t → ∞.

This property is an immediate consequence of Property 2 and the fact from calculus that a function which is bounded and monotone (either increasing or decreasing) must approach a limit as t → ∞. In order to show which solutions of a given differential equation have a limit, it is necessary to show which solutions remain bounded. A solution may become unbounded positively (i.e., y(t) → +∞) if it is monotone increasing when y is large. In many applications, g(y) < 0 for large y, and in such a case every solution remains bounded, because solutions which become large and positive must be decreasing. If, as is also frequently the case in applications, y = 0 is an equilibrium and only non-negative solutions are of interest, solutions cannot cross the line y = 0 and become negatively unbounded (i.e., y(t) → −∞). In many applications, y stands for a quantity such as the number of members of a population, the mass of a radioactive substance, the quantity of money in an account, or the height of a particle above ground which cannot become negative. In such a situation, only non-negative solutions are significant, and if y = 0 is not an equilibrium but a solution reaches the value zero for some finite t, we will consider the population system to have collapsed and the population to be zero for all larger t. If this is the case we need not be concerned with the possibility of solutions becoming negatively unbounded even if y = 0 is not an equilibrium. Property 4 The only possible limits as t → ∞ of solutions of the differential equation y ′ = g(y) are the equilibria of the differential equation. To see why this property is true, we use the fact from calculus that if a differentiable function tends to a limit as t → ∞, then its derivative must tend to zero. If a solution y(t) of y ′ = g(y) tends to zero, then by the continuity of the function g(y) we have 0 = lim y ′ (t) = lim g{y(t)} = g{ lim y(t)}. t→∞

t→∞

t→∞

Thus, limt→∞ y(t) must be a root of the equation g(y) = 0, and hence an equilibrium of (B.26). In applications, the initial condition usually comes from observations and is subject to experimental error. For a model to be a plausible predictor of what will actually occur, it is important that a small change in the initial value does not produce a large change in the solution. An equilibrium , y of (B.26) such that every solution with initial value sufficiently close to , y approaches , y as t → ∞ is said to be asymptotically stable. If there are solutions which start arbitrarily close to an equilibrium but move away from it, then the equilibrium is said to be unstable. For the logistic differential equation (B.8) the equilibrium y = 0 is unstable, and the equilibrium y = K is asymptotically stable. In applications, unstable equilibria

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Fig. B.5 Equilibria , y with (a) g ′ (, y ) < 0, (b) g ′ (, y) > 0

have no significance, because they can be observed only if the initial condition is “just right.” Property 5 An equilibrium , y of y ′ = g(y) with g ′ (, y ) < 0 is asymptotically ′ stable; an equilibrium , y with g (, y ) > 0 is unstable.

To establish this property, we note that if , y is an equilibrium with g ′ (, y) < 0 (g(y) has a negative slope at y = , y ), then g(y) is positive if y < , y and negative if y > , y (Fig. B.5a). In this case solutions above the equilibrium decrease toward the equilibrium, while solutions below the equilibrium increase toward the equilibrium. This shows that an equilibrium , y with g ′ (, y ) < 0 is asymptotically ′ stable. By a similar argument, if g (, y ) > 0, solutions above the equilibrium increase and solutions below the equilibrium decrease, with both moving away from the equilibrium (Fig. B.5b). The properties we have developed make it possible for us to make a complete analysis of the asymptotic behavior, that is, the behavior as t → ∞, of solutions of an autonomous differential equation (B.26) merely by examining the equilibria and the nature of the function g(y) for large values of y. We begin by finding all the equilibria (roots of the equation g(y) = 0). An equilibrium , y with g ′ (, y ) < 0 is asymptotically stable, and all solutions with initial value in the two bands adjoining this equilibrium tend to it. An equilibrium , y with g ′ (, y ) > 0 is unstable and repels all solutions with initial value in the two bands adjoining it. An equilibrium , y with g ′ (, y ) = 0 must be analyzed more carefully. If g(y) is negative for values of y above the largest equilibrium, then no solutions become positively unbounded. If g(y) is positive for values of y above the largest equilibrium, then this equilibrium is unstable, and solutions with initial value above this equilibrium become unbounded. If g(y) is negative for values of y below the smallest equilibrium, then this equilibrium is likewise unstable, and solutions with initial value below this equilibrium become negatively unbounded. A convenient way to display the qualitative behavior of solutions of an autonomous differential equation (B.26) is by drawing the phase line. We draw the graph of the function g(y) and on the y-axis we may draw arrows to the right where the graph is above the y-axis and to the left where the graph is below the y-axis. The reason for doing this is that where the graph is above the axis g(y) is positive and therefore the solution y of (B.26) is increasing, while where the graph is below the axis g(y) is negative and therefore the solution y of (B.26) is

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575

Fig. B.6 A phase line superimposed on the corresponding graph of g(y)

decreasing. The points where the graph crosses the y-axis are the equilibria, and we can see from the directions of the arrows along the axis which equilibria are asymptotically stable and which are unstable. The phase line is the y-axis viewed as the line on which the solution curve moves, thinking of t as a parameter. Thus, the solution is described by the motion along the line, whose direction is given by the arrows. The graph of Fig. B.6 describes a situation in which there are asymptotically stable equilibria at y = −1 and y = 2, and an unstable equilibrium at y = 0. Example 8 Describe the asymptotic behavior of solutions of the differential equation (B.25). Solution Here g(y) = −ay + b, g ′ (y) = −a. The only equilibrium is y = ab . If a > 0, this equilibrium is asymptotically stable, and g(y) < 0 if y is large and positive, g(y) > 0 if y is large and negative. This means that every solution is bounded and approaches the limit ab . If a < 0, however, the equilibrium is unstable. Further, since g(y) > 0 above the equilibrium and g(y) < 0 below the equilibrium, every solution is unbounded, either positively or negatively.  Example 9 Describe the asymptotic behavior of solutions with y(0) ≥ 0 of the differential equation

 y ′ = y re−y − d ,

where r and d are positive constants.

(B.27)

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Fig. B.7 A phase line and graph of g(y) for (B.27), with rd

Solution The equilibria of (B.27) are the solutions of y(re−y − d) = 0. Thus there are two equilibria, namely y = 0 and the solution , y of re−y = d, which is r , y = log d . If r < d, , y < 0, and only the equilibrium y = 0 is of interest. In this case, g(y) < 0 for y > 0, and solutions with y(0) > 0 decrease to 0 (Fig. B.7). If r > d, we define K to be log dr , so that the positive equilibrium is y = K.

 We may now rewrite the differential equation (B.27) as y ′ = ry e−y − e−K . For



 the function g(y) = ry e−y − e−K , we have g ′ (y) = r e−y − e−K − rye−y and g ′ (0) = r(1 − e−K ) > 0, implying that the equilibrium y = 0 is unstable. The equilibrium y = K is asymptotically stable since g ′ (K) = −rKe−K < 0. All positive solutions are bounded, because g(y) < 0 for y > K. Thus every solution with y(0) > 0 tends to the limit K, while the solution with initial value zero is the  zero function and has limit zero (Fig. B.8).

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We see that the solutions of (B.27), which has been suggested as a population growth model with per capita birth rate re−y and per capita death rate d, behave qualitatively in the same manner as solutions of the logistic equation. Some other differential equations which exhibit the same behavior have been proposed as population models. The original formulation of the logistic equation was based on a per capita growth rate which should be a decreasing function of population size, positive for small y but negative for large y. It is not difficult to show that every population model of the form (B.26) for which the per capita growth rate is a decreasing function of the population size y, and which is positive for 0 < y < K and negative for y > K, has the property that every solution with y(0) > 0 approaches the limit K as t → ∞. In many applications, the function g(y) has the form g(y) = yf (y), which has the effect of guaranteeing that y = 0 is an equilibrium. Then g ′ (y) = f (y)+yf ′ (y). At the equilibrium y = 0, g ′ (0) = f (0) and at a non-zero equilibrium , y with f (, y ) = 0, g ′ (, y) = , y f ′ (, y ). Thus the equilibrium y = 0 is asymptotically stable if f (0) < 0 and unstable if f (0) > 0; a non-zero equilibrium , y is asymptotically stable if f ′ (, y ) < 0 and unstable if f ′ (, y ) > 0. We shall restate this result formally as a theorem. Equilibrium Stability Theorem An equilibrium , y of y ′ = g(y) with g ′ (, y ) < 0 is ′ asymptotically stable; an equilibrium , y with g (, y ) > 0 is unstable. The equilibrium y = 0 of y ′ = yf (y) is asymptotically stable if f (0) < 0 and unstable if f (0) > 0, while a non-zero equilibrium , y is asymptotically stable if f ′ (, y ) < 0 and unstable if ′ f (, y ) > 0.

Appendix C

Systems of Differential Equations

In the previous chapter we saw how to model and analyze continuously changing quantities using differential equations. In many applications of interest there may be two or more interacting quantities—populations of two or more species, for instance, or parts of a whole, which depend upon each other. When the amount or size of one quantity depends in part on the amount of another, and vice versa, they are said to be coupled, and it is not possible or appropriate to model each one separately. In these cases we write models which consist of systems of differential equations. In this chapter, we will find that the quantitative and qualitative approaches we used to analyze individual differential equations in Sects. B.2 and B.5 extend in a more or less natural way to cover systems of differential equations. Extending them will require some basic multivariable calculus, principally the use of partial derivatives and the idea of linear approximation.

C.1 The Phase Plane Our purpose is to study two-dimensional autonomous systems of first-order differential equations of the general form y ′ = F (y, z)

(C.1)

′

z = G(y, z) Usually, it is not possible to solve such a system, by which we mean to find y and z as functions of t so that y ′ (t) = F (y(t), z(t)),

z′ (t) = G(y(t), z(t))

© Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9

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for all t in some interval. However, we can often obtain information about the relation between the functions y(t) and z(t). Geometrically, such information is displayed as a curve in the y–z plane, called the phase plane for this system An equilibrium of the system (C.1) is a solution (y∞ ,z∞ ) of the pair of equations F (y, z) = 0,

G(y, z) = 0.

(C.2)

Geometrically, an equilibrium is a point in the phase plane. In terms of the system (C.1), an equilibrium gives a constant solution y = y∞ , z = z∞ of the system. This definition is completely analogous to the definition of an equilibrium given for a first-order differential equation in Sect. B.5. The orbit of a solution y = y(t), z = z(t) of the system (C.1) is the curve in the y–z phase plane consisting of all points (y(t), z(t)) for 0 ≤ t < ∞. A closed orbit corresponds to a periodic solution because the orbit must travel repeatedly around the closed orbit as t increases. There is a geometric interpretation of orbits which is analogous to the interpretation given for solutions of first-order differential equations in Sect. B.3. Just as the curve y = y(t) has slope y ′ = f (t, y) at each point (t, y) along its length, an orbit of (C.1) (considering z as an implicit function of y) has slope dz G(y, z) z′ = ′ = dy y F (y, z) at each point of the orbit. The direction field for a two-dimensional autonomous system is a collection of line segments in the phase plane with this slope at each point (y, z), and an orbit must be a curve which is tangent to the direction field at each point of the curve. Computer algebra systems such as Maple and Mathematica may be used to draw the direction field for a given system. Example 1 Describe the orbits of the system y ′ = z,

z′ = −y.

Solution If we consider z as a function of y, we have dz y = z′ /y ′ = − . dy z Solution by separation of variables gives  2

2

z dz = −



y dy,

and integration gives z2 = − y2 + c. Thus every orbit is a circle y 2 + z2 = 2c with center at the origin, and every solution is periodic. 

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To find equilibria of a system (C.1), it is helpful to draw the nullclines, namely the curves F (y, z) = 0 on which y ′ = 0, and G(y, z) = 0 on which z′ = 0. An equilibrium is an intersection of these two curves.

C.2 Linearization of a System at an Equilibrium Sometimes it is possible to find the orbits in the phase plane of a system of differential equations, but it is rarely possible to solve a system of differential equations analytically. For this reason, our study of systems will concentrate on qualitative properties. The linearization of a system of differential equations at an equilibrium is a linear system with constant coefficients, whose solutions approximate the solutions of the original system near the equilibrium. In this section, we shall see how to find the linearization of a system. In the next section, we shall see how to solve linear systems with constant coefficients, and this will enable us to understand much of the behavior of solutions of a system near an equilibrium. Let (y∞ , z∞ ) be an equilibrium of a system (C.1) that is, a point in the phase plane such that (C.2). We will assume that the equilibrium is isolated, that is, that there is a circle centered around (y∞ , z∞ ) which does not contain any other equilibrium. We shift the origin to the equilibrium by letting y = y∞ + u, z = z∞ + v, and then make linear approximations to F (y∞ + u, z∞ + v) and G(y∞ + u, z∞ + v). The difference here between a one-dimensional system and a two-dimensional one is that the linear approximation in two or more dimensions uses partial derivatives. Our approximations are F (y∞ + u, z∞ + v) ≈ F (y∞ , z∞ ) + Fy (y∞ , z∞ )u + Fz (y∞ , z∞ )v

(C.3)

G(y∞ + u, z∞ + v) ≈ G(y∞ , z∞ ) + Gy (y∞ , z∞ )u + Gz (y∞ , z∞ )v with error terms h1 and h2 , respectively, which are negligible relative to the linear terms in (C.3) when u and v are small (i.e., close to the equilibrium). The linearization of the system (C.1) at the equilibrium (y∞ , z∞ ) is defined to be the linear system with constant coefficients u′ = Fy (y∞ , z∞ )u + Fz (y∞ , z∞ )v

(C.4)

v ′ = Gy (y∞ , z∞ )u + Gz (y∞ , z∞ )v.

To obtain it, we first note that y ′ = u′ , z′ = v ′ , and then substitute (C.3) into (C.1). By (C.2) the constant terms are zero, and for the linearization we neglect the higher-order terms h1 and h2 . The coefficient matrix of the linear system (C.4) is the matrix of constants   Fy (y∞ , z∞ ) Fz (y∞ , z∞ ) . Gy (y∞ , z∞ ) Gz (y∞ , z∞ )

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The entries of this Jacobian matrix describe the effect of a change in each variable on the growth rates of the two variables. Example 1 Find the linearization at each equilibrium of the system y ′ = A − βyz − μy,

z′ = βyz − (γ + μ)z.

(This corresponds to the classical Kermack–McKendrick SIR model for a possibly endemic disease. Here y corresponds to the number of susceptible, uninfected individuals, z to the number of infected, infective individuals, A to the birth rate, μ to the death rate and β and γ to the infection and recovery rates, respectively.) Solution The equilibria are the solutions of A = y(βz + μ), z(βy) = (γ + μ)z. To satisfy the second of these equations, we must have either z = 0 or βy = γ + μ. If z = 0, the first equation gives y = A/μ. If z > 0, the second equation gives βy = γ + μ. Since ∂ [−βyz − μy] = −βz − μ, ∂y ∂ [βyz − (γ + μ)z] = βz, ∂y

∂ [−βyz] = −βy, ∂z

∂ [βyz − (γ + μ)z] = βy − (γ + μ), ∂z

the linearization at an equilibrium (y∞ ,z∞ ) is u′ = −(βz∞ + μ)u − βy∞ v,

v ′ = βz∞ u + (βy∞ − (γ + μ))v.

At the equilibrium (A/μ, 0) the linearization is u′ = −μu − β v′ = β

A v μ

A − (γ + μ)v. μ

At the other equilibrium with I > 0 the linearization is u′ = −(βI∞ + μ)u − (μ + α)v v ′ = βI∞ u.



An equilibrium of the system (C.1) with the property that every orbit with initial value sufficiently close to the equilibrium remains close to the equilibrium for all t ≥ 0, and approaches the equilibrium as t → ∞, is said to be locally asymptotically stable. An equilibrium of (C.1) with the property that some solutions starting arbitrarily close to the equilibrium move away from it is said to be unstable.

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These definitions are completely analogous to those given in Sect. B.5 for first-order differential equations. We speak of local asymptotic stability to distinguish from global asymptotic stability, which is the property that all solutions, not merely those with initial value sufficiently close to the equilibrium, approach the equilibrium. If we speak of asymptotic stability of an equilibrium, we will mean local asymptotic stability unless we specify that the asymptotic stability is global. The fundamental property of the linearization which we will use to study stability of equilibria is the following result, which we state without proof. The proof may be found in any text which covers the qualitative study of nonlinear differential equations. Here we suppose F and G to be twice differentiable, that is, smooth enough for the linearization to give a correct picture. Linearization Theorem If (y∞ , z∞ ) is an equilibrium of the system y ′ = F (y, z),

z′ = G(y, z)

and if every solution of the linearization at this equilibrium approaches zero as t → ∞, then the equilibrium (y∞ , z∞ ) is (locally) asymptotically stable. If the linearization has unbounded solutions, then the equilibrium (y∞ , z∞ ) is unstable. For a first-order differential equation y ′ = g(y) at an equilibrium y∞ , the linearization is the first-order linear differential equation u′ = g ′ (y∞ )u. We may solve this differential equation by separation of variables and see that all solutions approach zero if g ′ (y∞ ) < 0, and there are unbounded solutions if g ′ (y∞ ) > 0. We have seen in Sect. B.5, without recourse to the linearization, that the equilibrium is locally asymptotically stable if g ′ (y∞ ) < 0 and unstable if g ′ (y∞ ) > 0. The linearization theorem is valid for systems of any dimension and is the approach needed for the study of stability of equilibria for systems of dimension higher than 1. Note that there is a case where the theorem above does not draw any conclusions, namely the case where the linearization about the equilibrium is neither asymptotically stable nor unstable. In this case, the equilibrium of the original (nonlinear) system may be asymptotically stable, unstable, or neither. Example 2 For each equilibrium of the system y ′ = z,

z′ = −2(y 2 − 1)z − y

determine whether the equilibrium is asymptotically stable or unstable. Solution The equilibria are the solutions of z = 0, −2(y 2 − 1)z − y = 0, and thus ∂ ∂ [z] = 0, ∂z [z] = 1, and the only equilibrium is (0,0). Since ∂y ∂ ∂ [−2(y 2 − 1)z − y] = −4yz − 1, [−2(y 2 − 1)z − y] = −2(y 2 − 1), ∂y ∂z

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the linearization at (0,0) is u′ = 0u + 1v = v, v ′ = −u + 2v. We can actually solve this system of equations outright by using a clever trick to reduce it to a single equation. We subtract the first equation from the second to give (v − u)′ = (v − u). This is a first-order differential equation for (v − u), whose solution is v − u = c1 et . This partial result is already enough to tell us that the equilibrium is unstable, as since the difference between u and v grows exponentially, at least one of them must therefore grow exponentially as well. As there are unbounded solutions, we conclude that the equilibrium (0,0) is unstable. 

C.3 Solution of Linear Systems with Constant Coefficients We have seen in the preceding section that the stability of an equilibrium of a system of differential equations is determined by the behavior of solutions of the system’s linearization at the equilibrium. This linearization is a linear system with constant coefficients (recall that a linear system has right-hand sides linear in y and z). Thus, in order to be able to decide whether an equilibrium is asymptotically stable, we need to be able to solve linear systems with constant coefficients. We were able to do this in the examples of the preceding section because the linearizations took a simple form with one of the equations of the system containing only a single variable. In this section we shall develop a more general technique. The problem we wish to solve is a general two-dimensional linear system with constant coefficients, y ′ = ay + bz,

(C.5)

z′ = cy + dz,

where a, b, c, and d are constants. We look for solutions of the form y = Y eλt , z = Zeλt ,

(C.6)

where λ, Y , and Z are constants to be determined, with Y and Z not both zero. When we substitute the form (C.6) into the system (C.5), using y ′ = λY eλt , z′ = λZeλt , we obtain two conditions λY eλt = aY eλt + bZeλt ,

λZeλt = cY eλt + dZeλt ,

which must be satisfied for all t. Because eλt = 0 for all t, we may divide these equations by eλt to obtain a system of two equations which do not depend on t, namely λY = aY + bZ, λZ = cY + dZ

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or (a − λ)Y + bZ = 0,

(C.7)

cY + (d − λ)Z = 0. The pair of equations (C.7) is a system of two homogeneous linear algebraic equations for the unknowns Y and Z. For certain values of the parameter λ this system will have a solution other than the obvious solution Y = 0, Z = 0. In order that the system (C.7) have a non-trivial solution for Y and Z, it is necessary that the determinant of the coefficient matrix, which is (a − λ)(d − λ) − bc, be equal to zero (this result comes from linear algebra). This gives a quadratic equation, called the characteristic equation, of the system (C.5) for λ. We may rewrite the characteristic equation as λ2 − (a + d)λ + (ad − bc) = 0.

(C.8)

We will assume that ad − bc = 0, which is equivalent to the assumption that λ = 0 is not a root of (C.8). Our reason for this assumption is the following: If λ = 0 is a root (or, equivalently, ad − bc = 0), the equilibrium conditions will reduce to a single equation since each equation is a constant multiple of the other and there is effectively only one equilibrium equation. Consequently, there will be a line of non-isolated equilibria. We do not wish to explore this problem in part because the treatment of non-isolated equilibria is complicated and in part because it is not possible to apply the linearization theorem of the previous section when the linearization of a system about an equilibrium is of this form. If ad − bc = 0, the characteristic equation has two roots λ1 and λ2 , which may be real and distinct, real and equal, or complex conjugates. If λ1 and λ2 are the roots of (C.8), then there is a solution (Y1 , Z1 ) of (C.7) corresponding to the root λ1 , and a solution (Y2 , Z2 ) of (C.7) corresponding to the root λ2 . These, in turn, give us two solutions y = Y1 eλ1 t ,

y = Y2 eλ2 t ,

z = Z1 eλ1 t

z = Z2 eλ2 t

of the system (C.5). We note that if λ is a root of (C.8), then equations (C.7) reduce to a single equation. Thus we may make an arbitrary choice for one of Y , Z and the other is then determined by (C.7). Because the system (C.5) is linear, it is easy to see that every constant multiple of a solution of (C.7) is also a solution and also that the sum of two solutions of (C.7) is also a solution. It is possible to show that if we have two different solutions of the system (C.5), then every solution of (C.5) is a constant multiple of the first solution plus a constant multiple of the second solution. By “different” we mean that neither solution is a constant multiple of the other. Here, if the roots λ1 and λ2 of the characteristic equation (C.8) are distinct, we do have two different solutions of (C.5), and then every solution of system (C.5) has the form

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y = K1 Y1 eλ1 t + K2 Y2 eλ2 t ,

(C.9)

z = K1 Z1 eλ1 t + K2 Z2 eλ2 t

for some constants K1 and K2 . The form (C.9) with two arbitrary constants K1 and K2 is called the general solution of the system (C.5). If initial values y(0) and z(0) are specified, these two initial values may be used to determine values for the constants K1 and K2 , and thus to obtain a particular solution in the family (C.9). Example 1 Find the general solution of the system y ′ = −y − 2z,

z′ = y − 4z

and also the solution such that y(0) = 3, z(0) = 1. Solution Here a = −1, b = −2, c = 1, d = −4, so that a + d = −5, ad − bc = 6, and the characteristic equation is λ2 + 5λ + 6 = 0, with roots λ1 = −2, λ2 = −3. With λ = −2, both equations of the algebraic system (C.7) are Y − 2Z = 0, and we may take Y = 2, Z = 1. The resulting solution of (C.5) is y = 2e−2t , z = e−2t . With λ = −3, both equations of the algebraic system (C.7) are Y − Z = 0, and we may take Y = 1, Z = 1. The resulting solution of (C.5) is y = e−3t , z = e−3t . Thus the general solution of the system is y = 2K1 e−2t + K2 e−3t ,

z = K1 e−2t + K2 e−3t .

To satisfy the initial conditions, we substitute t = 0, y = 3, z = 1 into this form, obtaining a pair of equations 2K1 + K2 = 3, K1 + K2 = 1. We may subtract the second of these from the first to give K1 = 2, and then K2 = −1. This gives, as the solution of the initial value problem, y = 4e−2t − e−3t ,

z = 2e−2t − e−3t . 

In the language of linear algebra, λ is an eigenvalue and (Y, Z) is a corresponding eigenvector of the 2 × 2 matrix A=



ab cd



Note that a +d is the trace, denoted by trA, and ad −bc is the determinant, denoted by detA, of the matrix A. The sum of the eigenvalues of A is the trace, and the product of the eigenvalues is the determinant. This is seen easily by writing (C.8) as λ2 − trAλ + detA = 0

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We will consider vectors to be column vectors, that is, matrices with two rows and one column. Suppose the matrix A has two distinct real eigenvalues λ1 , λ2 with corresponding eigenvectors [Y1 , Z1 ],

[Y2 , Z2 ]

respectively (this approach can also be used if λ1 = λ2 provided there are two independent corresponding eigenvectors). We define the matrix P =



Y1 Y2 Z1 Z2



and then if we make the change of variable        y u Y1 Y2 u =P = Z1 Z2 v z v the system (C.5) is transformed to the system u′ = λ1 u

(C.10)

v ′ = λ2 v.

Since the system (C.10) is uncoupled, it may easily be solved explicitly to give u = K1 eλ1 t ,

v = K2 eλ2 t .

We may then transform back to the original variables y, z to give the solution (C.9). The linear algebra approach gives the same result that was obtained by our initial approach of trying to find exponential solutions. Since it is constructive, it does not make use of the result which we stated without proof that if we have two different solutions of the system (C.5), then every solution of (C.5) is a constant multiple of the first solution plus a constant multiple of the second solution. If the characteristic equation (C.8) has a double root, the method we have used gives only one solution of the system (C.5), and we need to find a second solution in order to form the general solution. In order to see what form the second solution must have we consider the special case c = 0 of (C.5). Then (C.5) becomes y ′ = ay + bz,

(C.11)

′

z = dz. Then the eigenvalues of the corresponding matrix A are a and d; if a = d, the characteristic equation has a double root. However, the assumption c = 0 means that we can solve the system (C.11) recursively. Every solution of the second equation in (C.11) has the form

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z = Zedt and we may substitute this into the first equation to give y ′ = ay + bZedt This first order linear differential equation is easily solved. We multiply the equation by e−at to give y ′ e−at − aye−at = bZe(d−a)t .

(C.12)

If a = d, integration of (C.12) gives ye−at =

bZ (d−a)t e +Y d −a

where Y is a constant of integration. From this we obtain the solution y=

bZ dt e + Y eat d

z = Zedt which is equivalent to the solution (C.9) obtained earlier; note that here λ1 = a, λ2 = d. A more interesting case arises when a = d, so that the two roots of the characteristic equation are equal. Then (C.12) becomes y ′ e−at − aye−at = bZ

(C.13)

and integration gives ye−at = bZt + Y where Y is a constant of integration. From this we obtain the solution y = bZteat + Y eat z = Zeat .

This suggests that if there is a double root λ of the characteristic equation the needed second solution of (C.5) will contain terms teλt as well as eλt . It is possible to show (and the reader can verify) that if λ is a double root of (C.8), we must have λ = a+d 2

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and in addition to the solution y = Y1 eλt , z = Z1 eλt of (C.5) there is a second solution, of the form y = (Y2 + Y1 t)eλt ,

z = (Z2 + Z1 t)eλt

where Y1 , Z1 are as in (C.7) and Y2 , Z2 are given by (a − λ)Y2 + bZ2 = Y1 , cY2 + (d − λ)Z2 = Z1 . Thus the general solution of the system (C.5) in the case of equal roots is y = (K1 Y1 + K2 Y2 )eλt + K2 Y1 teλt , z = (K1 Z1 + K2 Z2

)eλt

+ K2 Z1

(C.14)

teλt .

Note that if (C.8) has a single root and b = 0, then λ = a = d, and we have Y1 = 0, Z1 = cY2 , so that the general solution becomes y = K3 eλt , z = K4 eλt + cK3 teλt , where K3 ≡ K2 Y2 and K4 ≡ cK1 Y2 + K2 Z2 are arbitrary constants. Likewise if (C.8) has a single root and c = 0, then λ = a = d, Z1 = 0, Y1 = bZ2 , and the general solution reduces to y = K5 eλt + bK6 teλt ,

z = K6 eλt

with arbitrary constants K5 ≡ bK1 Z2 + K2 Y2 and K6 ≡ K2 Z2 . If b and c are both zero, the system becomes uncoupled y ′ = ay,

z′ = dz

and is easily solved by integration to give y = K1 eat , z = K2 edt . This is the solution in all cases, regardless of whether the characteristic equation has distinct roots or equal roots. Example 2 Find the general solution of the system y ′ = z,

z′ = −y + 2z

and also the solution such that y(0) = 2, z(0) = 3. Solution Since a = 0, b = 1, c = −1, d = 2, we have a + d = 2, ad − bc = 1. The characteristic equation is λ2 − 2λ + 1 = 0, with a double root λ = 1. With λ = 1, both equations of the system (C.7) are Y − Z = 0, and we may take Y = 1, Z = 1 to give the solution y = et , z = et of (C.5). Substituting these values into the

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equations for Y2 , Z2 , we find that they reduce to the single equation −Y2 + Z2 = 1, so we may take Y2 = 0, Z2 = 1. Equations (C.14) now give the general solution y = K1 et + K2 tet , z = (K1 + K2 )et + K2 tet . To find the solution with y(0) = 2, z(0) = 3, we substitute t = 0, y = 2, z = 3 into this form, obtaining the pair of equations K1 = 2, K1 + K2 = 3. Then K2 = 1, and the solution satisfying the initial conditions is y = 2et + tet , z = 3et + tet .  From a linear algebra perspective we may make a change of variable of the system (C.5) just as in the case where we had two eigenvectors. We define the matrix 

Y Y P = 1 2 Z1 Z2



with 

Y1 Z1



an eigenvector of the matrix A as before, but with 

Y2 Z2



chosen arbitrarily, except that the determinant of P must be non-zero. Then under the change of variable        y u u Y Y =P = 1 2 Z1 Z2 v z v the system (C.5) is transformed to a system of the form (C.5) with c = 0 which may be solved explicitly, as we have seen. A particularly clever choice of [Y2 , Z2 ] is the solution of the system of linear equations aY2 + bZ2 = λY2 + Y1 cY2 + dZ2 = λZ2 + Y2 . It is possible to show that this system always has a solution. With this choice of Y2 , Z2 , the system (C.5) is transformed to u′ = λ1 u + v ′

v = λ2 v.

(C.15)

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If we solve the system (C.15) recursively by the method used to solve the system (C.11) and the solution is then transformed back to the original variables, we obtain the solution (C.14). Another complication arises if the characteristic equation (C.8) has complex roots. While the general solution of (C.5) is still given by (C.9), in this case the constants λ1 and λ2 are complex, and the solution is in terms of complex functions. Complex exponentials, however, can be defined with the aid of trigonometric functions (eiθ ≡ cos θ + i sin θ for real θ ), so it is still possible to give the solution of (C.5) in terms of real exponential and trigonometric functions. In this case, if the characteristic equation (C.8) has conjugate complex roots λ = α ± iβ, where α and β are real and β > 0, equations (C.9) become y = (K1 Y1 + K2 Y2 )eαt cos βt + i(K1 Y1 − K2 Y2 )eαt sin βt,

z = (K1 Z1 + K2 Z2 )eαt cos βt + i(K1 Z1 − K2 Z2 )eαt sin βt. We can eliminate the imaginary coefficients by defining Q1 ≡ i(K1 − K2 ), Q2 ≡ K1 + K2 , and taking (a − λi )Yi + bZi = 0 for i = 1, 2 from (C.9) to arrive at the form y = Q1 beαt sin βt + Q2 beαt cos βt,

z = −[Q1 (a − α) − Q2 β]eαt sin βt + [Q1 β − Q2 (a − α)]eαt cos βt.

Example 3 Find the general solution of the system y ′ = −2z,

z′ = y + 2z

and also the solution with y(0) = −2, z(0) = 0. Solution We have a = 0, b = −2, c = 1, d = 2, and the characteristic equation is λ2 − 2λ + 2 = 0, with roots λ = 1 ± i. The general solution is then y = −2K1 et sin t − 2K2 et cos t,

z = (K1 − K2 )et sin t + (K1 + K2 )et cos t.

To find the solution with y(0) = −2, z(0) = 0, we substitute t = 0, y = −2, z = 0 into this form, obtaining −2K2 = −2, K1 + K2 = 0, whose solution is K1 = −1, K2 = 1. This gives the particular solution y = 2et sin t − 2et cos t,

z = −2et sin t. 

In many applications, especially in analyzing stability of an equilibrium, the precise form of the solution of a linear system is less important to us than the qualitative behavior of solutions. It will turn out that often the crucial question is whether all solutions of a linear system approach zero as t → ∞. The nature of the origin as an equilibrium of the linear system (C.5) depends on the roots of the

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characteristic equation (C.8). If both roots of (C.8) are real and negative, then the solutions of (C.5) are combinations of negative exponentials. This implies that all orbits approach the origin and the origin is asymptotically stable. If the roots of (C.8) are real and of opposite sign, then there are solutions which are positive exponentials and solutions which are negative exponentials. Thus there are solutions approaching the origin and solutions moving away from the origin, and the origin is unstable. If the roots of (C.8) are complex, the orbits approach the origin if their real parts are negative. We conclude that the origin is an asymptotically stable equilibrium of (C.5) if both roots of (C.8) have negative real part and unstable if at least one root of(C.8) has positive real part. Solution of systems of linear differential equations with constant coefficients can be carried out more economically with the use of vectors and matrices. A system y1′ = a11 x1 + a12 x2 + . . . + a1n xn

y2′ = a21 x1 + a22 x2 + . . . + a2n xn .. .. .. . . .

yn′ = am1 x1 + am2 x2 + . . . + amn xn may be written in the form y ′ = Ay,

(C.16)

where A is an n × n matrix and y and y ′ are column vectors, ⎡

a11 ⎢a21 ⎢ A=⎢ . ⎣ ..

a12 a22 .. .

an1 an2

⎤ . . . a1n . . . a2n ⎥ ⎥ . . .. ⎥ . . ⎦ . . . ann ,

⎡ ⎤ y1 ⎢y2 ⎥ ⎢ ⎥ y = ⎢ . ⎥. ⎣ .. ⎦ yn

We attempt to find solutions of the form y = eλt c with c a constant column vector. Substituting this form into (C.16) we obtain the condition λeλt c = Aeλt c, or Ac = λc. Thus λ be an eigenvalue of the matrix A and c must be a corresponding eigenvector. If the eigenvalues of A are distinct, this procedure generates n solutions of (C.16) and it can be shown that every solution is a linear combination of these solutions. This explains the exponential solutions that we obtained. If there are multiple

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eigenvalues of A, it may be necessary to obtain additional solutions, and these will be exponential functions multiplied by powers of t. If some eigenvalues are complex, the corresponding complex exponential solutions may be replaced by products of exponential functions and sines or cosines.

C.4 Stability of Equilibria In order to apply the linearization theorem of Sect. C.2 to questions of stability of an equilibrium, we must determine conditions under which all solutions of a linear system with constant coefficients y ′ = ay + bz, z′

(C.17)

= cy + dz

approach zero as t → ∞. As we saw in Sect. C.3, the nature of the solutions of (C.17) is determined by the roots of the characteristic equation λ2 − (a + d)λ + (ad − bc) = 0.

(C.18)

If the roots λ1 and λ2 of (C.18) are real, then the solutions of (C.17) are made up of terms eλ1 t and eλ2 t , or eλ1 t and teλ1 t if the roots are equal. In order that all solutions of (C.17) approach zero, we require λ1 < 0 and λ2 < 0, so that the terms will be negative exponentials. If the roots are complex conjugates, λ = α ±iβ, then in order that all solutions of (C.17) approach zero, we require α < 0. Thus if the roots of the characteristic equation have negative real part, all solutions of the system (C.17) approach zero as t→ ∞. In a similar manner, we may see that if a root of the characteristic equation has positive real part, then (C.17) has unbounded solutions. It turns out, however, that it is not necessary to solve the characteristic equation in order to determine whether all solutions of (C.17) approach zero, as there is a useful criterion in terms of the coefficients of the characteristic equation. The basic result is that the roots of a quadratic equation λ2 +a1 λ+a2 = 0 have negative real part if and only if a1 > 0 and a2 > 0. Applying this to the characteristic equation (C.18) and the system (C.17), we obtain the following result for linear systems with constant coefficients: Stability Theorem for Linear Systems Every solution of the linear system with constant coefficients (C.17) approaches zero as t → ∞ if and only if the trace a + d of the coefficient matrix of the system is negative and the determinant ad − bc of the system’s coefficient matrix is positive. If either the trace is positive or the determinant is negative, there is at least one unbounded solution.

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Example 1 Determine whether all solutions tend to zero or whether there are unbounded solutions for each of the following systems: (i) y’ = -y - 2z, z’ = y - 4z (ii) y’ = z, z’ = -u - 2v (iii) y’ = -2z, z’ = y + 2z Solution (i) The characteristic equation is λ2 +5λ+6 = 0, with roots λ = −2, λ = −3. Thus all solutions tend to zero. Alternatively, since the trace of the coefficient matrix is −5 < 0 and the determinant is 6 > 0, the stability theorem gives the same conclusion. For (ii), the characteristic equation is λ2 + 2λ + 1 = 0 with a double root λ = −1, and thus all solutions tend to zero. For (iii), the characteristic equation is λ2 − 2λ + 2 = 0, and since the trace is positive, there are unbounded solutions. As we indicated in the previous section, we could also have drawn this conclusion from a phase portrait.  If we apply the stability theorem for linear systems to the linearization (C.4) of a system (C.2)at an equilibrium (y∞ , z∞ ), we obtain the following result. Equilibrium Stability Theorem Let (y∞ , z∞ ) be an equilibrium of a system y ′ = F (y, z), z′ = G(y, z), with F and G twice differentiable. Then if the trace of the Jacobian matrix at the equilibrium is negative and the determinant of the Jacobian matrix at the equilibrium is positive, the equilibrium (y∞ , z∞ ) is locally asymptotically stable. If either the trace is positive or the determinant is negative, then the equilibrium (y∞ , z∞ ) is unstable. Example 2 Determine whether each equilibrium of the system y ′ = z,

z′ = 2(y 2 − 1)z − y

is locally asymptotically stable or unstable. Solution The equilibria are the solutions of z = 0, 2(y 2 − 1)z − y = 0, and thus the only equilibrium is (0,0). Here F (y, z) = z, with partial derivatives 0 and 1, respectively, and G(y, z) = 2(y 2 − 1)z − y, with partial derivatives 4yz − 1 and 2(y 2 − 1), respectively. Therefore the Jacobian matrix at the equilibrium is 

0 1 −1 −2



with trace −2 and determinant 1, as in Example 1(ii). Thus the equilibrium (0,0) is locally asymptotically stable. 

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Example 3 Determine whether each equilibrium of the system y ′ = y(1 − 2y − z) z′ = z(1 − y − 2z)

is locally asymptotically stable or unstable. Solution The equilibria are the solutions of y(1 − 2y − z) = 0, z(1 − y − 2z) = 0. One solution is (0,0); a second is the solution of y = 0, 1 − y − 2z = 0, which is (0, 12 ); a third is the solution of z = 0, 1 − 2y − z = 0, which is ( 12 ,0); and a fourth is the solution of 1 − 2y − z = 0, 1 − y − 2z = 0, which is ( 13 , 13 ). The Jacobian matrix at an equilibrium (y∞ , z∞ ) is 

 1 − 4y∞ − z∞ −y∞ . −z∞ 1 − y∞ − 4z∞

At (0,0), this matrix has trace 1 and determinant 1, and thus the equilibrium is unstable. At (0, 12 ), this matrix has trace − 21 and determinant − 12 , and thus the equilibrium is unstable. At ( 21 ,0), this matrix has trace − 12 and determinant − 21 , and thus the equilibrium is unstable. At ( 13 , 31 ), this matrix has trace − 34 and determinant 1  3 , and thus this equilibrium is locally asymptotically stable. The careful reader will have noticed that, like the linearization theorem of Sect. C.2, the equilibrium stability theorem given above has a hole of sorts in its result, in that the theorem says nothing about the stability of equilibria for which the trace and determinant lie on the boundary of conditions, in other words, for which the trace or the determinant of the Jacobian matrix at the equilibrium is zero, so that the linearization has solutions which do not approach zero as t → ∞ but stay bounded. The reason for this “hole” is that in such cases, the linearization does not give enough information to determine stability. If all orbits beginning near an equilibrium remain near the equilibrium for t ≥ 0, but some orbits do not approach the equilibrium as t → ∞, the equilibrium is said to be stable, or sometimes neutrally stable. If the origin is neutrally stable for the linearization at an equilibrium, then the equilibrium may also be neutrally stable for the nonlinear system. However, it is also possible for the origin to be neutrally stable for the linearization at an equilibrium, while the equilibrium is asymptotically stable or unstable. Thus neutral stability of the origin for a linearization at an equilibrium gives no information about the stability of the equilibrium. We have seen in Sect. B.5 that a solution of an autonomous first-order differential equation is either unbounded or approaches a limit as t → ∞. For an autonomous system of two first-order differential equations, these same two possibilities exist. In addition, however, there is the possibility of an orbit which is a closed curve, corresponding to a periodic solution. Such an orbit is called a periodic orbit because it is traversed repeatedly.

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Fig. C.1 Trajectories approaching a limit cycle

There is a remarkable result which says essentially that these are the only possibilities. Poincaré-Bendixson Theorem A bounded orbit of a system of two first-order differential equations which does not approach an equilibrium as t→ ∞ either is a periodic orbit or approaches a periodic orbit as t→ ∞. It is possible to show that a periodic orbit must enclose an equilibrium point in its interior. In many examples, there is an unstable equilibrium, and orbits beginning near this equilibrium spiral out towards a periodic orbit. A periodic orbit which is approached by other (non-periodic) orbits is called a limit cycle. One example of a limit cycle involves the system y ′ = y(1 − y 2 − z2 ) − z,

z′ = z(1 − y 2 − z2 ) + y,

which has its only equilibrium at the origin. The equilibrium is unstable, and Fig. C.1 illustrates the fact that all orbits not beginning at the origin spiral counterclockwise [in or out] toward the unit circle y 2 + z2 = 1. In many applications the functions y(t) and z(t) are restricted by the nature of the problem to non-negative values. For example, this is the case if y(t) and z(t) are population sizes. In such a case, only the first quadrant y ≥ 0, z ≥ 0 of the phase plane is of interest. For a system y ′ = F (y, z), z′ = G(y, z) which has F (0, z) ≥ 0 for z ≥ 0 and G(y, 0) ≥ 0 for y ≥ 0, then since y ′ ≥ 0 along the positive z-axis (where y = 0) and z′ ≥ 0 along the positive y-axis (where z = 0), no orbit can leave the first quadrant by crossing one of the axes. The Poincaré– Bendixson theorem may then be applied to orbits in the first quadrant. In such a case, the first quadrant is called an invariant set, a region with the property that orbits must remain in the region. If instead F and G are identically zero along the respective [half-]axes, then y ′ = 0 for {y = 0, z ≥ 0}, and z′ = 0 for {y ≥ 0, z = 0}. In this case, orbits which

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begin on an axis must remain on that axis, and orbits beginning in the interior of the first quadrant (with y(0) > 0, z(0) > 0) must remain in the interior of the first quadrant (i.e., y(t) > 0 and z(t) > 0 for t ≥ 0). If there is no equilibrium in the first quadrant, there cannot be a periodic orbit, because a periodic orbit must enclose an equilibrium. Thus, if there is no equilibrium in the first quadrant every orbit must be unbounded. Example 4 Show that every orbit in the region y > 0, z > 0 of the system y ′ = y(2 − y) −

yz y+1 ,

yz −z z′ = 4 y+1

approaches a periodic orbit as t → ∞.

Solution We have y ′ = 0 when y = 0, and z′ = 0 when z = 0, so orbits starting in the first quadrant remain in the first quadrant. Equilibria are the solutions of either 4y z y = 0 or 2 − y = y+1 , and either z = 0 or y+1 = 1. If y = 0, we must also have z = 0. If 2 − y = 20 9 .

z y+1 ,

we could have z = 0, which implies y = 2, or y = 31 , which

implies z = Thus there are three equilibria, namely (0,0), (2,0), and ( 13 , 20 9 ). By checking the values of the trace and determinant of the Jacobian matrix, which is 

2 − 2y∞ −

z∞ (1+y∞ )2

4z∞ (y∞ +1)2

y∞ − y∞ +1

4y∞ y∞ +1

−1



,

we may see that each of the three equilibria is unstable. In order to apply the Poincaré–Bendixson theorem, we must show that every orbit starting in the first quadrant of the phase plane is bounded. To show this, we might like to show that y ′ and z′ are negative when y and/or z are sufficiently large, but a glance at the equations tells us this isn’t necessarily so. Therefore we instead consider some positive combination of y and z whose time derivative does become negative far enough from the origin. In particular, consider the function V (y, z) = 4y + z. If an orbit is unbounded, then along this orbit the function V (y, z) must also be unbounded. The derivative of V (y, z) along an orbit is d V [y(t), z(t)] = 4y ′ (t) + z′ (t) = 4y(2 − y) − z. dt This is negative except in the bounded region defined by the inequality z < 4y(2 − y). Therefore the function V (y, z) cannot become unbounded, because it is decreasing (dV /dt < 0) whenever it becomes large (z > 4y(2 − y), which is true, for example, whenever V > 9). This proves that all orbits of the system are bounded. Now we may apply the Poincaré–Bendixson theorem to see that every orbit approaches a limit cycle. 

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C.5 Qualitative Behavior of Solutions of Linear Systems In Sect. C.2 we reduced the analysis of the stability of an equilibrium (x∞ , y∞ ) of a system of differential equations x ′ = F (x, y),

y ′ = G(x, y)

to the determination of the behavior of solutions of the linearization at the equilibrium u′ = Fx (x∞ , y∞ )u + Fy (x∞ , y∞ )v,

(C.19)

v ′ = Gx (x∞ , y∞ )u + Gy (x∞ , y∞ )v.

Next we shall analyze the various possibilities for the behavior of solutions of the two-dimensional linear homogeneous system with constant coefficients x ′ = ax + by,

(C.20)

y ′ = cx + dy,

where a, b, c, and d are constants, in order that we might describe the behavior of solutions of the linearization at an equilibrium. We will then be able to state some refinements of the linearization theorem of Sect. C.3 that give more specific information about the behavior of solutions near an equilibrium as determined by the Jacobian matrix at the equilibrium. We will assume throughout that ad − bc = 0. This implies that the origin is the only equilibrium of the system (C.20). If this system is the linearization at an equilibrium (x∞ , y∞ ) of a nonlinear system (C.19), then this equilibrium is isolated, meaning that there is a disc centered at (x∞ , y∞ ) containing no other equilibrium of the nonlinear system. It is convenient to use vector–matrix notation: We let x denote the column vector xy , x′ the column vector  x′  y ′ , and A the 2 × 2 matrix   ab . cd

Then, using the properties of matrix multiplication, we may rewrite the linear system x ′ = ax + by, y ′ = cx + dy in the form x′ = Ax. A linear change of variable x = P u, with P a nonsingular 2 × 2 matrix (which represents a rotation of the axes and a change of scale along the axes), transforms the system to P u′ = AP u, or u′ = P −1 AP u = Bu.

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This is of the same type as the original system x′ = Ax, and its coefficient matrix B = P −1 AP is similar to A. If we can solve for u, we can then reconstruct x = P u, and in fact the qualitative properties of solutions for u are preserved in this reconstruction. Thus, we may describe the various possible phase portraits of x′ = Ax by listing the various possible canonical forms of A under similarity and constructing the phase portrait for each possibility. Canonical Form Theorem The matrix   ab A= , cd with detA = ad − bc = 0, is similar under a real transformation to one of   λ0 (i) , λ > μ > 0 or λ < μ < 0. 0μ   λ0 (ii) , λ > 0 or λ < 0. 0λ   λ1 (iii) , λ > 0 or λ < 0. 0λ   λ0 (iv) , λ>0>μ 0μ   0 β (v) , β = 0. −β 0   α β (vi) , α > 0, β = 0 or α < 0, β = 0. −β α We now describe the phase portraits for each of these cases in turn. Case (i) The transformed system is u′ = λu, v ′ = μv, with solution u = u0 eλt , v = v0 eμt . If λ < μ < 0 then u and v both tend to zero as t → ∞ and v/u = v0 e(μ−λ)t /u0 → +∞. Thus, every orbit tends to the origin with infinite slope (except if v0 = 0, in which case the orbit is on the u-axis), and the phase portrait is as shown in Fig. C.2. If λ > μ > 0 the portrait is the same, except that the arrows are reversed. Case (ii) The system is u′ = λu, v ′ = λv, with solution u = u0 eλt , v = v0 eλt . If λ < 0 both u and v tend to zero as t → ∞ and v/u = v0 /u0 . Thus every orbit is a straight line going to the origin, and all slopes as the orbit approaches the origin are possible. The phase portrait is as shown in Fig. C.3. If λ > 0, the portrait is the same except that the arrows are reversed. Case (iii) The system is u′ = λu + v, v ′ = λv. The solution of the second equation is v = v0 eλt , and substitution into the first equation gives the first-order linear equation u′ = λu + v0 eλt whose solution is u = (u0 + v0 t)eλt . If λ < 0, both u and v tend to zero as t → ∞, and v/u = v0 /(u0 + v0 t) tends to zero unless v0 = 0, in

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Fig. C.2 Case (i)

Fig. C.3 Case (ii)

which case the orbit is the u- axis. Because u = (u0 + v0 t)eλt , we have du/dt =

 λt (u0 λ + v0 ) + λv0 t e , and therefore du/dt = 0 when t = −(v0 + u0 λ)/λv0 . Thus except for the orbits on the u-axis, every orbit has a maximum or minimum u-value and then turns back toward the origin. The phase portrait is as shown in Fig. C.4. If λ > 0, the portrait is the same except that the arrows are reversed. Case (iv) The solution is u = u0 eλt , v = v0 eμt just as in Case (i), but now u is unbounded and v → 0 as t → ∞, unless u0 = 0, in which case the orbit is on the v-axis. The phase portrait is as shown in Fig. C.5. Case (v) The system is u′ = βv, v ′ = −βu. Then u′′ = βv ′ = −β 2 u and u = A cos βt + B sin βt for some A, B. Thus v = u′ /β = −A sin βt + B cos βt and u2 +v 2 = A2 +B 2 . Every orbit is a circle, clockwise if β > 0 and counterclockwise if β < 0. The phase portrait for β > 0 is as shown in Fig. C.6.

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Fig. C.4 Case (iii)

Fig. C.5 Case (iv)

Fig. C.6 Case (v)

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Fig. C.7 Case (vi)

Case (vi) The system is u′ = αu + βv, v ′ = −βu + αv. We make the change of variables u = eαt p, v = eαt q, so that u′ = αeαt p + eαt p′ , v ′ = αeαt q + eαt q ′ reduces the system to p′ = βq, q ′ = −βp, which has been solved in Case (v). Thus, u = eαt (A cos βt + B sin βt), v = eαt (−A sin βt + B cos βt), and u2 + v 2 = e2αt (A2 + B 2 ). If α < 0, u2 + v 2 decreases exponentially, and the orbits are spirals inward to the origin, clockwise if β > 0 and counterclockwise if β < 0. The phase portrait is as shown in Fig. C.7. If α > 0, the portraits are the same except that the arrows are reversed. These six cases may be classified as being of four distinct types. In Cases (i), (ii), and (iii) all orbits approach the origin as t → +∞ (or as t → −∞ depending on the signs of λ and μ) with a limiting direction, and the origin is said to be a node of the system. In Case (iv), only two orbits approach the origin as t → +∞ or as t → −∞, and all other orbits move away from the origin. In this case the origin is said to be a saddle point. In Case (vi) every orbit winds around the origin in the sense that its angular argument tends to +∞ or to −∞, and the origin is said to be a vortex, spiral point, or focus. In Case (v), every orbit is periodic; in this case the origin is said to be a center. According to the equilibrium stability theorem of Sect. C.4, asymptotic stability of the origin for the linearization implies asymptotic stability of an equilibrium of a nonlinear system. In addition, instability of the origin for the linearization implies instability of an equilibrium of a nonlinear system. The asymptotic stability or instability of the origin for a linear system is determined by the eigenvalues of the matrix A, defined to be the roots of the characteristic equation 

 a−λ b det(A − λI ) = det = (a − λ)(d − λ) − bc c d −λ = λ2 − (a + d)λ + (ad − bc) = 0.

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The sum of the eigenvalues is the trace of the matrix A, namely a + d, and the product of the eigenvalues is the determinant of the matrix A, namely ad − bc. A similarity transformation preserves the trace and determinant and therefore does not change the eigenvalues. Thus, the eigenvalues of A are λ, μ in Cases (i) and (iv); λ (a double eigenvalue ) in Cases (ii) and (iii); the complex conjugates ±iβ in Case (v); and α ± iβ in Case (vi). Examination of the phase portraits in the various cases shows that the origin is asymptotically stable in Case (i) if λ < μ < 0; in Cases (ii) and (iii) if λ < 0; and in Case (vi) if α < 0. Similarly the origin is unstable in Case (i) if λ > μ > 0; in Cases (ii) and (iii) if λ > 0; in Case (iv); and in Case (vi) if α > 0. The origin is stable but not asymptotically stable in Case (v) (center). A simpler description is that the origin is asymptotically stable if both eigenvalues have negative real part and unstable if at least one eigenvalue has positive real part. If both eigenvalues have real part zero, the origin is stable but not asymptotically stable. Our assumption that ad − bc = det A = 0 rules out the possibility that λ = 0 is an eigenvalue; thus, eigenvalues with real part zero can occur only if the eigenvalues are pure imaginary, as in Case (v). Combining this analysis with the linearization result, we have the following result (which is the equilibrium stability theorem of Sect. C.4 with some added details): Stability Theorem If (x∞ , y∞ ) is an equilibrium of the system (C.19) and if all eigenvalues of the coefficient matrix of the linearization at this equilibrium have negative real part, specifically if trA(x∞ , y∞ ) = Fx (x∞ , y∞ ) + Gy (x∞ , y∞ ) < 0, detA(x∞ , y∞ ) = Fx (x∞ , y∞ )Gy (x∞ , y∞ ) − Fy (x∞ , y∞ )Gx (x∞ , y∞ ) > 0, then the equilibrium (x∞ , y∞ ) is asymptotically stable. We can be more specific about the nature of the orbits near an equilibrium. In terms of the elements of the matrix A, we may characterize the cases as follows, using the remark that the eigenvalues are complex if and only if Δ = (a + d)2 − 4(ad − bc) = (a − d)2 + 4bc < 0. 1. If detA = ad − bc < 0, the origin is a saddle point. 2. If detA > 0 and trA = a + d < 0, the origin is asymptotically stable, a node if Δ ≥ 0 and a spiral point if Δ < 0. 3. If det A > 0 and trA > 0, the origin is unstable, a node if Δ ≥ 0 and a spiral point of Δ > 0. 4. If detA > 0 and trA = 0, the origin is a center. It is possible to show that in general, the phase portrait of a nonlinear system at an equilibrium is similar to the phase portrait of the linearization at the equilibrium, except possibly if the linearization has a center. This is true under the assumption that the functions F (x, y) and G(x, y) in the system (C.19) are smooth enough that Taylor’s theorem is applicable, so that the terms neglected in the linearization

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process are of higher order. If the linearization at an equilibrium has a node, then the equilibrium of the nonlinear system is also a node, defined to mean that every orbit tends to the equilibrium (either as t → ∞ or as t → −∞) with a limiting direction. If the linearization at an equilibrium has a spiral point, then the equilibrium of the nonlinear system is also a spiral point, defined to mean that every orbit tends to the equilibrium (either as t → ∞ or as t → −∞) with its angular variable becoming infinite. If the linearization at an equilibrium has a saddle point, then the equilibrium of the nonlinear system is also a saddle point. A saddle point is defined by the characterization that there is a curve through the equilibrium such that orbits starting on this curve tend to the equilibrium but orbits starting off this curve cannot stay near the equilibrium. An equivalent formulation is that there are two orbits tending to the equilibrium as t → +∞ and two orbits tending away from the equilibrium, or tending to the equilibrium as t → −∞. These orbits are called separatrices; the two orbits tending to the saddle point are the stable separatrices, while the two orbits tending away from the saddle point are the unstable separatrices. Other orbits appear like hyperbolas (Fig. C.8). A center is defined to be an equilibrium for which there is an infinite sequence of periodic orbits around the equilibrium with the orbits approaching this equilibrium. If the linearization at an equilibrium has a center, then the equilibrium of the nonlinear system may be a center, but is not necessarily a center: It could be an asymptotically stable spiral point or an unstable spiral point. Fig. C.8 Separatrices at a saddle point

Index

A Absorbing boundary, 466 Accelerated progression, 267 Acquired immunity, 394 Acquired resistance, 251 Active dynamics, 274 Activity level, 189, 190, 210, 211, 477 Adjoint equation, 258 Adjoint problem, 258 Adverse effect of control, 161 Aedes aegypti, 409, 410, 414 Age, 14 cohort, 69 at infection, 69, 71 of infection, 6, 10, 37, 139–141, 143, 145, 149, 153, 155, 156, 168–171, 175, 195, 209, 210, 269, 273, 281, 347, 452, 464 profile, 331 structure, 15, 265, 344, 429 Age dependent birth rate, 449 Agent-based model, 362, 384 Agents bacterial, 23 helminth, 23 Age progression rate, 447 Age specific disease induced death rate, 447 Age specific natural death rate, 447 Age specific recovery rate, 447 Age-structured models, 15 Age-structured populations, 15 AIDS, 10, 23, 273–276, 279, 282, 283, 286, 302, 303 Airline network, 461 Airline travel, 461

Air travel, 457 Algebraic equation, 539, 585 Algorithm, 549 Amplitude, 34, 71, 72 Antibiotic, 250 resistance, 256, 511 resistant strain, 256 Antigenically distinct strain, 347 Antigenic drift, 323 Antigenic shift, 326 Antimalarial therapy, 405 Antimicrobial resistance, 511 Antiviral drug, 159, 312, 319, 322, 511 Antiviral efficacy, 322 Antiviral treatment, 221, 312, 319, 320, 322, 334, 510, 511 Approximate exponential growth, 356 Arbitrary constant, 567 Arbitrary distribution, 376 Arbitrary stage distribution, 367 Arbitrary stage duration distribution, 162, 163 Arbovirus, 414 Aristotle, 5 Assessment, 338 Asymptomatic, 11, 319, 347, 413, 414, 419 case, 315 compartment, 313 infection, 313 infective, 217 period, 312, 314 stage, 133, 195 Asymptotically stable, 573, 582 Asymptotic behavior, 26, 574 Asymptotic spread, 464

© Springer Science+Business Media, LLC, part of Springer Nature 2019 F. Brauer et al., Mathematical Models in Epidemiology, Texts in Applied Mathematics 69, https://doi.org/10.1007/978-1-4939-9828-9

605

606 Asymptotic stability, 11, 12, 26, 29, 31–33, 49, 51, 55, 63, 64, 67, 68, 74, 75, 77–82, 87, 88, 101, 109, 113, 130, 182, 183, 185, 186, 223, 234, 238, 243, 254, 255, 264, 265, 269, 278, 284, 288, 304, 339, 340, 344, 345, 393, 400, 402, 404, 406, 436, 444, 445, 449, 450, 459, 460, 468, 471, 472, 520, 573–577, 582–584, 592, 594, 595, 602, 603 global, 64, 583 local, 583 Attack rate, 37, 314, 315, 318 symptomatic, 323 Attack ratio, 195 Augmented matrix, 544 Autonomous, 563, 570, 571, 579, 580, 595

B Backward bifurcation, 263, 265, 511 Bacteria, 22 Balance equation, 222, 465 Balance relation, 234 Baseline value, 315 Basic reproduction number, 6, 8, 10–12, 14, 22, 26, 28, 33, 39, 44, 53, 63, 65, 68, 71, 74, 78, 83, 112, 117, 120, 123, 125, 132, 135, 140, 141, 146, 148, 151–153, 156, 168, 169, 171, 180, 182, 184, 187, 191, 192, 196, 211– 214, 223, 232, 234, 235, 243, 275, 277, 280, 283, 304, 305, 313–315, 317, 322, 325, 327, 339, 344, 347, 348, 351, 352, 361, 380, 385, 387, 392–396, 400, 406, 411–417, 422, 449, 450, 452, 453, 460, 478, 479, 484, 489, 492, 496–498 Behavior, 161 Behavioral change, 14, 384 Behavioral class, 29 Behavioral heterogeneity, 429 Behavioral response, 334 Bernouilli, 5 Berra, Yogi, 508 Between-host system, 108, 109 Bifurcation, 81, 338, 460 backward, 12, 81, 82, 85–87 curve, 12, 81, 87, 88 diagram, 341, 403, 404 forward, 12, 81, 82 parameter, 275, 339, 403 point, 340, 341 transcritical, 12

Index Bilinear incidence, 43 Birth defects, 13, 418 Birth modulus, 437 Birth rate, 100, 404, 479 density dependent, 31, 400 maximum, 403 Bi-stability, 404 Bite, 231 Black Death, 3, 337 Blood meal, 229 Boosting of immunity, 396, 397 Boundary condition, 448, 466 Boundary equilibrium, 254, 255, 288 asymptotically stable, 344 Boundary layer, 238, 239, 243 Branching process, 7, 8, 118, 129 Broad Street pump, 5 Bubonic plague, 41, 100, 102 Budd William, 5 Burial, 360

C Carbon dating, 557 Carrier, 406, 407 Carrying capacity, 28, 72, 82 Case, 352 data, 336 fatality rate, 385 fatality ratio, 45 Case-finding, 251, 257 Case-holding, 251, 257, 260 Cell mortality, 108 Cell–parasite interaction, 107 Census data, 102 Center, 602, 604 manifold, 341 Chagas’ disease, 13, 67, 229 Chain binomial model, 8 Changing behavior, 274 Characteristic equation, 33, 75, 76, 130, 133, 137, 233, 352, 417, 436, 549, 585, 587–589, 591–594, 602 Characteristics, 437 Chemoprophylaxis, 262 Chicken pox, 15, 22, 71, 429 Chikungunya, 13, 229, 414 Cholera, 3, 5, 11, 21, 107, 167 Clinical attack rate, 315 Clinical case, 314, 315 Coefficient matrix, 32, 33 Coevolutionary interaction, 346 Coexistence, 254, 256, 345, 348 equilibrium, 254

Index multiple strain, 347 region, 254, 345 steady state, 269 sub-threshold, 347 Cohort, 25 Coinfection, 266, 267, 284, 334, 347, 510 Communicable disease, 478 Compartment, 117 Compartmental diagram, 284 Compartmental model, 6–9, 11, 22–24, 26, 35, 81, 137, 141, 182, 189, 195, 312, 331 Compartmental structure, 23, 26, 64, 144, 155, 230 Competing disease strains, 49 Competing risks, 4 Competitive exclusion principle, 347 Compliance, 250, 252 Computer network, 7, 118 Conditional probability, 304 Condom use, 273 Configuration model, 126 Conservation law, 465 Constant coefficients, 569, 570, 581, 584, 592, 593 Constant solution, 566, 567, 570, 571, 580 Constraint, 222, 266, 367, 378, 449 Contact, 7, 44, 118, 131, 139, 161, 188, 189, 196, 201, 216, 219, 276, 281, 286, 304, 313, 316, 319, 404, 460, 477 effective, 24 intervention, 125 matrix, 196 network, 7, 8, 118, 123, 124, 312 pattern, 118 random, 23, 24 tracing, 134, 216, 510 Contact rate, 15, 31, 39, 69, 72, 151, 156, 189, 201, 204, 208, 210, 212, 216, 221, 242, 274, 322, 332, 333, 385, 386, 458, 486 pandemic, 327 variable, 332 Contact tracing, 360, 361 Contagion, 509 Contagious agent, 337 Continuous time model, 46 Control, 159, 188, 259, 318, 380, 485, 491 effort, 256 measure, 9, 14, 97, 134–136, 151, 165, 166, 188, 315, 351, 360, 381, 490, 494, 509 parameter, 97, 136, 148 program, 259, 263, 351, 500

607 reproduction number, 135, 148, 163, 165, 166, 180, 188, 196, 197, 199, 261, 287, 317, 322, 324, 379, 380, 382 strategy, 97, 216, 386 Cordon sanitaire, 386, 387, 481, 483, 484 Core group, 14, 188 Cost, 338 Coupled processes, 400 Coupling strength, 458 Critical mass, 509 Critical point, 346 Critical threshold, 394 Critical transmissibility, 125 Cross-immunity, 11, 111, 264, 323–328, 338, 344–347, 409, 510 group, 326 model, 324

D Damping, 72 Data, 362 Death, 364, 365, 372, 379 modulus, 437 rate, 100, 282, 404, 405 natural, 72 per capita, 403 Decay rate, 553 Degree distribution, 118, 122, 124, 126 Delta function, 201, 202, 466 Demographic effects, 6, 27, 117, 312, 511 Demographic equation, 51 Demographic equilibrium, 51 Demographic model, 459 Demographic parameter, 400 Demographic process, 28 Demographics, 25, 63, 230 Demographic steady state, 277 Demographic time scale, 72 Dengue, 229, 230, 409, 414, 421, 491, 501 fever, 13 hemorrhagic fever, 409 pathogenicity, 409 shock syndrome, 409 spread, 422 Density dependence, 436 Density-dependent birth rate, 29 Determinant, 32, 33, 67, 74, 88, 133, 188, 546, 585, 590, 593–595, 603 Deterministic model, 8, 23, 42, 122, 362 Diagnosis rate, 418 Diagonalization, 551 Diagonal matrix, 543, 551 Diarrhea, 3

608 Difference equation, 121 Differential–difference equation, 10, 24, 75 Differential equation, 10, 24 Differential infectivity, 134 Diffusion, 467, 468, 471, 509 coefficient, 465, 469 instability, 471 Diffusivity, 465 Diphtheria, 71 Direct contact rate, 419 Direct host to host transmission, 414 Direction field, 561, 562, 580 Direct transmission, 414 number, 415 Discrete model, 217 Discrete time system, 47 Disease case, 321 disease case, 331 Disease compartment, 182–184 Disease control, 337, 338 Disease death, 31, 43, 315, 322 Disease death rate, 44, 45, 100, 314, 469 Disease debilitation, 21 Disease dynamics, 407 Disease-free equilibrium, 27 Disease-induced death, 359 Disease-induced mortality, 363 Disease intervention, 96 Disease mortality, 21, 100, 276, 380, 405 Disease outbreak, 7, 23, 319, 334, 386 Disease prevalence, 97, 252, 289, 406, 478, 511 Disease progression, 182 Disease progression rate, 448 Disease stage, 218 duration, 89 Disease state, 187 Disease strain, 510 Disease survival rate, 314 Disease symptoms, 315 Disease transmission, 7, 14, 15, 166, 217, 273, 275, 351, 394, 418, 477 sexual, 418 Dispersion coefficients, 52 Distribution of stay, 117, 168, 209 Distribution time of waiting time, 365 Domain of attraction, 81, 88 Drinking, 274 Drug, 360 injection, 286 resistance, 334, 510, 511 treatment, 252, 256 compliance, 256 use, 273 Drug-resistant strain, 11, 251, 252

Index Drug-sensitive strain, 251, 252, 266, 270 Duration, 365, 381 Dynamic behavior, 275, 365 Dynamics of DENV-2, 422

E Early detection, 358 Early diagnosis, 359 Early outbreak data, 382 Ebola, 6, 351, 352, 356, 359, 362, 380, 382–384, 386, 387, 463, 480, 481, 484, 501 control, 358 Ecology, 398 Economic loss, 338 Edge, 7, 8, 118, 459 Effective contact, 131, 140, 142, 144 Effective human population size, 242 Effective population size, 485, 492 Effective reproduction number, 204, 206, 224, 351, 356, 359 Effective sub-population, 204 Efficacy, 337 Eigenvalue, 32, 67, 71, 75, 84, 101, 130, 133, 137, 184–186, 190, 191, 194, 205, 215, 232, 233, 324, 400, 401, 406, 416, 417, 463, 473, 548, 587, 592, 602, 603 dominant, 353 double, 603 Eigenvector, 184, 185, 215, 587, 590, 592 Endemic, 21, 22, 230, 241, 249, 287, 332, 394 disease, 11, 21, 26 Environmental contamination, 107, 108 Environmental risk, 477, 478 Epidemic, 6, 14, 22, 23, 36, 44, 73, 117, 122, 123, 133, 140, 159, 179, 211, 216, 230, 312, 318, 331–333, 459, 471 control, 359 curve, 381 death rate, 45 duration, 173 final size, 117, 159, 160, 164, 165, 173, 174, 212, 214, 216, 352, 354, 381, 419, 481–484, 494–498, 500, 511 growth, 383 outbreak, 351 peak, 331 size, 159, 160 time, 159, 160 recurrent, 102, 347 recurring, 71 of rumors, 56

Index size, 419 spread, 457 threshold, 359 wave, 330, 332, 509 Epidemic model, 539 discrete-time, 162 with general exposed and infectious periods, 146 with general quarantine/isolation, 146 with quarantine and isolation, 146, 162 with time-dependent infection rate, 159 Epidemiological class, 368 Epidemiological data, 102 Epidemiological model, 539 Epidemiological time scale, 72 Equilibrium, 11, 26, 31–33, 36, 63, 72, 100, 109, 112, 121, 130, 133, 234, 253, 280, 283, 344, 384, 393, 458, 468, 571–575, 580–585, 591, 594–596 age distribution, 438 analysis, 26 asymptotically stable, 32–34, 64, 67, 68, 73, 77, 79, 82, 88, 101, 183, 185, 186, 269, 277, 278, 284, 402, 449, 459, 468, 472, 573, 583, 592 asymptotically stable boundary, 404 asymptotically stable disease-free, 234, 243, 265, 288, 304, 339, 400 asymptotically stable endemic, 12, 234, 243, 265, 281, 304, 340 asymptotically stable interior, 288, 404 boundary, 112, 345 coexistence, 290 disease-free, 7, 11, 32, 63, 64, 66, 68, 74–76, 79, 82, 84, 91, 130, 136, 182–186, 217, 223, 233, 264, 275, 277, 278, 284, 324, 352, 385, 392, 393, 417, 449, 450, 453, 458, 460 endemic, 7, 11, 28, 32–34, 63, 66, 67, 70, 71, 73–75, 77, 79, 84, 87, 91, 95, 101, 223, 252, 264, 265, 275, 278, 284, 339, 340, 392, 393, 401, 452, 458, 511 globally asymptotically stable, 478, 479 globally stable disease-free, 478 interior, 288–290 isolated, 581, 598 locally asymptotically stable, 595 mosquito population, 410 positive, 29, 265, 281 stable, 340, 345, 595 stable boundary endemic, 345 stable endemic, 263 stable non-trivial, 344

609 stability theorem, 577, 594, 595 temporary, 356 unique asymptotically stable interior, 404 unique disease-free, 338 unstable, 77, 186, 278, 284, 288, 340, 573, 583, 595, 596 unstable endemic, 78, 81 unstable interior, 404 Estimated rate of exponential rise, 358 Estimated rate of rise of exponential fit, 358 Estimation of initial growth, 351 Eulerian perspective, 386, 461, 509 EVD, 351, 354, 356–358, 360, 386 Evolution, 398 Excess degree, 119–121 distribution, 124 Existence and uniqueness theorem, 560, 572 Exogenous reinfection, 263, 264 Experimental data, 334 Explicit movement, 458 Exponential distribution, 88, 90, 93, 99, 137, 150, 152, 158, 218, 367, 369–372, 376, 379 Exponential growth, 30, 101, 102, 353, 356, 382, 385, 386, 555 Exponentially distributed infective period, 146, 153 Exponentially distributed period, 145, 156, 158 Exponentially distributed stage, 167 Exponentially distributed waiting time, 276 Exponential rise, 357, 358 of epidemic curve, 356 Exponential waiting time, 364 distribution, 278 Exposed, 23, 37, 40, 64, 89–93, 95, 104, 129, 131, 135, 141, 142, 144, 146, 148, 154–156, 158, 159, 162–164, 170, 172–174, 182, 186, 189, 217, 230, 242, 275, 330, 337, 338, 347, 352, 361, 362, 395, 409, 420, 447, 448, 478, 481 period, 189 period distribution, 155 Extinction, 252, 460

F Fall wave, 335, 336 peak, 336 Family of functions, 558 Family of solutions, 555 Farr, William, 5 Fast dynamics, 400 Fast progression, 261, 268

610 Fast system, 402 Fast time scale, 400, 406 Fast variable, 108 Fatality rate, 330 Fick’s law, 465, 466 Final compartment size, 326 Final epidemic size, 174 Final size, 241, 347, 348, 387 of an heterogeneous mixing epidemic, 212 equation, 318, 321 inequality, 45 relation, 10, 37, 39, 40, 132, 134, 137, 139, 141, 180, 181, 186, 194, 195, 212–214, 216, 313, 314, 318, 325, 327, 328, 347, 387, 422, 509 First wave, 330–332 Fitness, 266, 403, 404 Flavivirus, 409 Flow diagram, 313, 317, 343 Flux, 465 density, 465 Focus, 602 Force of infection, 196, 202, 221, 252, 397, 458 Fox population, 469 Fox rabies, 464 Functional equation, 10, 24, 437, 439 Fundamental theorem of calculus, 566

G Galton–Watson, 7 Gamma distribution, 88–90, 92, 93, 95, 98, 99, 148–150, 152, 154, 156, 159, 167, 218, 367, 369, 372, 374, 376, 377 Garki project, 394 Gauss–Jordan method, 544, 546 Gaussian kernel, 202 Gene, 402, 403 General distribution, 374 of stay, 141 Generalized growth model, 382, 383 General stage distribution, 166, 369 Generating function, 119, 120, 122–124, 126 moments, 119 Generation time, 14, 201 Genetic composition, 400 Genetic variation, 323 Genotype, 399, 400 Geometric distribution, 165, 217, 218 Gladwell, 509 Global asymptotic stability, 26, 33, 53, 73, 277, 278, 339, 404, 478, 479, 520 Globally asymptotically stable, 277 Global spread, 490

Index Gonorrhea, 22, 23 Graph, 7, 118, 119, 457, 459, 572 Great Plague, 3, 40 Growth rate, 553, 560 Guillin–Barré syndrome, 414

H HAART, 273 Half-life, 556 Hamiltonian, 257 Hazard of infection, 126 Heat equation, 466 Hepatitis, 67 Herd immunity, 68, 346 Heterogeneity, 14, 129, 162, 189, 204, 208, 212, 213, 217, 219, 223, 224, 265, 266, 275, 347, 477, 490 of mixing, 161 Heterogeneous contact rates, 477 Heterogeneous mixing, 14, 65, 131, 188, 202, 209, 212–214, 216, 219, 274, 347, 429, 510 Heterosexual transmission, 274 Heterozygote, 398 H5N1 influenza epidemic, 6 High mobility, 484 High risk, 221, 487, 490–492, 497 area, 481, 482 region, 488 HIV/AIDS, 11, 14, 21, 67, 230, 249, 266, 267, 273–276, 279, 284, 286, 291, 302, 382 dynamics, 274 incidence, 286 infections new, 286 secondary, 287 infective individuals, 286 prevalence, 287 sexually transmitted, 274 transmission, 273, 286 vertical transmission, 286 Homoclinic bifurcation, 341, 342 Homoclinic loop, 404 Homoclinic orbit, 342 Homogeneity, 265 Homogeneous, 221 mixing, 7, 14, 118, 161, 209, 210, 212–214, 286 Homosexual population, homogeneously mixing, 281 Homosexual transmission, 273 Homozygote, 398

Index H1N1 influenza epidemic, 6 H1N1 strain, 311 Hopf bifurcation, 77, 78, 284, 339–341, 344 Hospitalization, 360, 363–365, 367, 372, 379–381 Host, 13, 229, 230, 346, 391 asymptomatically infected, 405 genetic structure, 398 infective, 235 population, 234, 393, 449, 494, 496 population size, 394 Host–vector system, 500 Human host, 448 Hygiene, 161, 216 Hyperbola, 604 I IDEA model, 382, 384 Identity matrix, 544, 549 Immigration, 102, 183, 404 Immune, 447 Immune response, 39 Immunity, 4, 7, 23, 29, 44, 63, 73, 251, 284, 336 against reinfection, 22, 394 average period, 397 boosting, 11 loss, 397 mean period, 397 permanent, 405 temporary, 23, 50, 73, 75, 396 Immunization, 23, 39, 68 program, 208 Immunological change, 326 Immunological experience, 346 Immunological ideas, 10 Immunological model, 284 Immunology, 5 Incidence data, 361 time series points, 358 Incidence function, 98 Incidence rate, 339 Incomplete treatment, 270 Incubation period, 274, 312, 352, 354, 357 arbitrary, 304 Incubation period distribution, exponential, 275 Indefinite integral, 565 Index case, 184, 338 Infection, 319, 485 age, 270, 281 age density, 281 age-dependent progression, 263 asymptomatic, 405–407

611 level, 291 repeated, 405 symptomatic, 406, 407 transmission, 490 Infection period, fixed length, 396 Infection rate, 14, 242, 331, 448, 462, 485 seasonally forced, 337 Infectious agent, 337 Infectiousness, 347 Infectious period distribution, 140 Infective, 5, 7, 8, 10–12, 14, 21–31, 33–35, 37, 39–41, 43–46, 48–50, 52, 55, 58, 63–69, 72, 73, 75, 77, 78, 81, 82, 89, 90, 93, 100–102, 104, 107, 108, 117, 118, 122, 124–127, 129–135, 137, 138, 141, 144–147, 149, 155–157, 159, 162, 164, 167–170, 173–175, 179–182, 184, 189, 190, 209, 215–217, 219, 230–232, 235, 242, 244, 251, 252, 264, 268, 269, 273, 275–280, 284, 287, 302–304, 312–315, 319, 321, 322, 332, 338, 352, 359, 360, 362–364, 367, 384, 386, 388, 392, 393, 410, 411, 413, 421, 439, 440, 447, 448, 458–462, 469, 472–474, 481, 510, 582 compartment, 313 stage, 189 Infective period, 25, 72, 137, 139, 146, 150, 156, 158, 183, 199, 219, 221, 274, 315, 319, 322, 335, 357, 458 distribution, 144, 155, 158, 214, 216, 218 fixed length, 146 Infectivity, 10, 11, 65, 131, 135, 139, 141, 144, 148, 168, 169, 190, 195, 209–211, 273, 282, 312, 313, 319, 323, 384 distribution, 141 Influenza, 11, 22, 65, 131, 133, 154, 195, 313, 314, 346, 501, 511 avian, 311 cases, 53 H1N1, 311, 331 H1N1 strain, 330, 334 H1N1 subtype, 323, 326 H2N2 subtype, 326 H3N2 subtype, 323, 326 pandemic, 312, 327, 328, 330–332, 334, 461 seasonal, 179, 328, 330, 332 seasonal epidemic, 316, 323, 332 seasonal outbreak, 326 subtype, 323 two strain, 338 virus interaction, 347

612 Infrectivity, 316 In-hospital transmission, 162, 217 Inital exponential growth, 352 Initial boundary value problem, 466 Initial condition, 190, 313, 317, 440, 448, 449, 466, 467, 554, 555, 559, 573 Initial distribution, 440 Initial exponential growth rate, 14, 39, 130, 131, 133, 137, 140, 141, 143, 148, 151, 153, 155, 156, 211, 212, 214, 233, 417, 418 Initial growth, 354 Initial growth rate, 353 exponential, 10 Initial infective distribution, 140 Initial per capita growth, 402 Initial population size, 404 Initial secondary vertex, 121 Initial size, 347, 348 Initial stochastic phase, 382 Initial time, 335, 354 Initial value, 571 problem, 554, 555, 559, 566 Inoculation, 71 Instability, 602 Integral equation, 10, 24, 280, 367, 374, 452 Integration by parts, 25 Integro-differential equation, 263, 365, 367, 376 Interaction, 196 Intercohort mixing, 440 Interepidemic period, 72 Interpolation, 102 Intervention, 346, 352, 354 effectiveness, 338 measures, 275 program, 489 strategy, 354 Intracohort mixing, 440 Invasion, 460 criterion, 140, 211 Inverse matrix, 545, 548 Isolated, 359 mixing, 206, 208 Isolation, 23, 37, 89, 93, 99, 111, 134–136, 146–148, 162, 165–167, 173, 174, 216–218, 337–341, 343, 344, 346, 347, 359–361, 388, 510, 512 Ivermectin, 241–243

J Jacobian, 217 matrix, 32, 66, 352, 401, 582, 594, 595, 598

Index Japanese encephalitis, 13, 229 Joint dynamics, 284

K Kermack–McKendrick model, 6, 7, 10, 26, 35, 73, 100, 117, 118, 128, 129, 139, 151, 152, 582 Kronecker delta, 196, 201

L Lagrangian approach, 477, 478, 481, 490 Lagrangian framework, 477, 486 Lagrangian model, 480, 501 Lagrangian modeling perspective, 480 Lagrangian perspective, 386, 421, 461, 477 Laplace transform, 148 Laplacian, 471 Latency period, 249 Latent, 249, 250 class, 275 Latent period, 195, 363 distribution, 263, 268 Latent stage, exponentially distributed, 269 Legrand model, 361–367, 372, 378, 379 Leishmaniasis, 13, 229 Life expectancy, 4, 69, 70, 72 Life span, 72 Like-with-like mixing, 192 Limit cycle, 596 Limiting system, 281 Linear approximation, 581 Linear chain trick, 92, 218 Linearization, 11, 31–33, 36, 49, 63, 66, 68, 71, 75, 76, 84, 87, 101, 130, 133, 136, 182, 183, 186, 233, 356, 385, 417, 581–585, 594, 595, 598, 602, 603 theorem, 583, 593 theory, 384 Linear system, 539 of equations, 539, 544 Line of equilibria, 36 Lipschitz structure, 258 Logistic equation, 27, 560, 561, 568–571, 577 Logistic model, 383 Long range connection, 123 Loss of immunity, 397 Lotka characteristic equation, 434 Lotka–Volterra model, 101 Low mobility, 484 Low risk, 221, 491, 492 area, 481 region, 483, 488

Index Lyme disease, 13, 229 Lymphatic filariasis, 13, 229

M Major epidemic, 8, 118, 120, 122, 123, 125 Malaria, 3, 4, 6, 11, 13, 15, 21, 22, 229, 230, 391, 393, 397, 403, 407, 429, 447 asymptomatic, 405, 407 control, 6, 394 disease dynamics, 400 endemic, 394 falciparum, 398 incubation period, 394 outbreak, 394 prevalence, 407 recovery time, 399 transmission, 400 Management model, 137 Management strategy, 311, 312, 315, 382 Maple, 562, 580 Mass action, 6, 122, 167, 234, 283, 404, 458 incidence, 24, 26, 43 Mathematica, 562, 580 Matlab, 562 Matrix, 133, 136, 137, 181, 183, 184, 187, 190, 194, 232, 233, 539, 581, 592, 595 algebra, 539 Jacobian, 186 non-negative, 185 operation, 542 product, 542 Maximum, 40 epidemic size, 213, 214 number of infectives, 41 McKendrick equation, 438 Mean degree, 119, 126 Mean duration, 366 Mean excess degree, 120 Mean exposed period, 158 Mean generation time, 156–158 Mean incubation period, 276 Mean infective period, 25, 33, 44, 72, 104, 131, 138, 140, 146, 150, 153, 158, 166, 167, 172, 189, 275, 278, 280, 303, 361, 405 death-adjusted, 277, 280, 304 symptomatic, 405 Mean infectivity, 139, 155, 156 Mean number of sexual partners, 276–278, 304 Mean sexual life expectancy, 276, 303 Mean sojourn time, 91, 95 Mean symptomatic attack rate, 315

613 Mean transmissibility, 124 Mean transmission rate, 359 Measles, 3, 4, 15, 21, 68, 71, 72, 429, 461 vaccine, 4 Mechanistic model, 382, 383 Medicated class, 242 Medicated host, 242 Meningitis, 22 Metapopulation, 197, 202, 204, 208, 223, 384, 457–461, 509 Michaelis–Menten interaction, 43 Microcephaly, 414, 490 Microorganisms, 22 Migration, 386, 461, 509 Mild dengue fever, 409 Minor outbreak, 8, 118, 122, 125 Mixing, 191, 199, 208, 210, 223, 265, 266 function, 201, 202, 204 matrix, 204, 206 pattern, 14, 188, 199, 200 M-matrix, 185 non-singular, 186 Mobile, 497 Mobility, 477, 480, 483–486, 488–500 behavior, 478 level, 484 matrix, 496 pattern, 500 regime, 500 restriction, 481 Model of dengue fever, 409 for Ebola, 351 endemic diseases, 63 linear age structured, 430 nonlinear age structured, 437 outcome, 365 prediction, 381 with quarantine, 174 Ross–MacDonald, 393 of Zika virus, 419 Modeling assumption, 385 Modeling framework, 338 Morbidity, 311, 337 Mortality, 311, 337 constant per capita, 398 fraction, 45 rate, 14, 278, 398 Mosquito, 6, 391, 394 biting rate, 393, 397 control, 409 control program, 397 repellent, 414 Movement restriction, 386

614 Multidrug-resistance, 250 Multi-patch model, 480 Multiple epidemic waves, 331 Multiple pandemic waves, 334 Multiple waves, 332 Multivariable calculus, 579 Mutation, 11, 73

N Natural death, 459 Natural death rate, 100, 251, 448, 479 Natural mortality, 276 Neighbor, 127 Net growth rate, 465 Network model, 8, 188, 189 Network structure, 331 Next generation approach, 463 Next generation matrix, 182, 187, 188, 190, 193, 194, 197, 204, 210, 215, 217, 218, 232, 313, 317, 324, 387, 392, 395, 400, 406, 412, 413, 415–417, 458, 460 with large domain, 184, 186–188, 193, 415 1918 influenza epidemic, 3 Nobel prize, 230 Nobel Prize in Medicine, 394 Node, 126, 128, 602–604 Non-carrier, 406, 407 Non-disease compartment, 182 Normal form, 341 Nosocomial transmission, 510 Nullcline, 581

O Occupied edge, 124 Offspring, 434 Onchocerciasis, 13, 229, 241 Onset, 364, 365, 367 Optimal allocation, 209 Optimal control, 251, 257–260, 266 Optimal strategy, 266 Optimal treatment, 258 Optimal treatment rate, 511 Optimal vaccination strategy, 223, 266 Optimal vaccine allocation, 205 Optimal vaccine policy, 205 Optimization, 205 Orbit, 37, 102, 468, 580, 582, 595, 596 Oscillation, 34, 71, 72, 77, 93, 332, 345 damped, 71, 458 Oscillatory solutions, 284

Index Outbreak, 72, 122, 140, 179, 356 Overall sojourn, 379

P Pair formation, 14, 43, 429, 445 Pandemic, 214, 216, 311, 312, 315, 322, 323, 326–329, 331, 332, 334, 336, 511 influenza, 326, 347 1918 influenza, 326 reproduction number, 327 strain, 311, 326, 327, 329, 330, 348 Parameter, 335 Parasite, 107, 346 Parasite load, 108 Parasitemia, 399 Partial derivative, 581 Partial differential equation, 436, 464 Partial fractions, 568 Partial immunity, 323 Particle, 465 concentration, 465 Partition, 436 Partner interaction, 14 Patch, 52, 386, 421, 457–461, 464, 471, 472, 477, 478, 480, 483, 485, 486, 489–492, 494–498, 500 coupling, 458 density, 500 residence time, 477 Patch-specific reproduction number, 479, 480 Patch-specific risk, 491 Pathogen, 11, 107, 108, 161, 167–171, 510 Pathogenicity, 34 Patient zero, 8, 119, 120 Pattern in space, 467 Peak size, 381, 382 Peak time, 336 Pearson chi-square statistics, 335 Period, 71 Periodic orbit, 77, 596 stable, 243 Periodic solution, 49, 339–342, 344, 345, 404, 580, 595 Period of stay, 152, 158 Perron–Frobenius theorem, 205 Perturbation technique, 344 Perturbed system, 341 Phase line, 571, 574 Phase locking, 458 Phase plane, 580 Phase portrait, 41, 468, 594, 599, 600, 602, 603

Index Phenomenological model, 382, 383 Piecewise exponential fit, 357, 358 Piecewise exponential rise, 357 Plague, 13, 229 Plasmodium, 399 Poincaré–Bendixson theorem, 596, 597 Poisson distribution, 50, 122, 123, 165 Policy decision, 322 Policy recommendation, 9 Polynomial equation, 33 Pontryagin maximum principle, 257 Population density, 33, 304, 399, 465, 497 ecology, 466 growth, 100, 102 homosexually active, 275 size, 189 subgroup, 331 Population model, 577 age structured, 429 Power series, 119 Pre-epidemic treatment, 317 Preferential mixing, 196, 200, 223 Preferred mixing, 191, 192, 204 Premedication class, 242 Pre-symptomatic case detection, 359 Pre-symptomatic infection, 359 Prevailing subtype, 326 Prevalence, 199, 406, 486, 488, 490 Prevalent, 291 Primary infected vertex, 122 Primary infection, 264 Primary resistance, 251 Prior immunity, 344 Probability, 8 density, 142 distribution, 122, 369 Progression, 485 to active tuberculosis, 290 to AIDS, 281, 290 rate, 242, 252 speed, 469 Properly posed problem, 31 Prophylaxis, 256, 261 Proportional death rate, 28, 29 Proportionate mixing, 191, 192, 195, 204, 206, 208, 211–213, 223, 445, 446 Public health infrastructure, 501 Public health measure, 161, 216, 334 Public health official, 386 Public health physicians, 6, 21 Public health policy, 312 Public transportation, 477

615 Q Qualitative analysis, 31, 101, 570 Qualitative behavior, 29, 35, 44, 394, 568, 574, 591, 598 Qualitative property, 569, 570 Quantitative framework, 338 Quarantine, 21, 22, 37, 40, 42, 89, 93, 95, 98, 99, 134–136, 146, 148, 162, 164–167, 173, 174, 216, 337–341, 343, 354, 510, 512, 523 Quarantine–isolation model, 175 Quasi-steady-state, 234, 237, 239, 243

R Rabies, 469, 471 Radioactive decay, 555 Random mixing, 446 Random partnership, 445 Rate of flow, 24, 465 Rate of transfer, 24 Reaction-diffusion, 464 Reaction-diffusion equation, 466–468 Reactivation, 488 Recovered, 447, 481 compartment, 313 Recovery, 359, 364, 365, 367, 372, 388 Recovery rate, 10, 44, 45, 202, 324, 377, 388, 479, 480 exponentially distributed, 35 Recruitment, 29, 281 Recruitment rate, 276, 278 Reduced system, 237–239 Reed–Frost model, 8 Region of asymptotic stability, 77 Region of attraction, 404 Reinfection, 22, 263, 266, 394, 488–490, 511 potential, 486 rate, 485 Relapse rate, 319 Relative infectivity, 324 Relative prevalence, 406, 407 Relative susceptibility, 407 Relative transmissibility, 359, 388 Removal rate, 278, 279, 305 variable, 278 Removed, 10, 23, 31, 35, 39, 73, 90, 100, 189, 196, 230, 312, 313, 352, 359, 362, 386, 388, 396, 421, 439, 461 Renewal condition, 440 Renewal equation, 433 Replacement, 348

616 Reproduction number, 96, 97, 125, 148, 154, 159, 166, 180–182, 185, 186, 190, 194, 197, 204, 205, 207, 208, 212, 213, 217–220, 222, 252–254, 262–264, 266, 269, 274, 275, 278, 287, 288, 291, 312, 318, 328, 329, 347, 354, 361, 374, 378, 387, 406, 417, 418, 420, 439, 452, 481, 509, 510 HIV, 287 with gamma distributed periods, 159 with general infectious period distribution, 153 stage-specific, 164 Reproductive rate, 553 Residence matrix, 478 Residence time, 461, 478, 481, 484–486, 494, 496, 498, 500, 509 Residence time matrix, 478 Resident, 386, 459 Resident population size, 461 Resistance, 266 Resistant strain, 252, 253, 256, 266, 269–271 Respiratory infection, 3, 334 Reversion from immune to susceptible, 396 Richards model, 382, 383 Rift Valley fever, 13, 229 Rinderpest, 67, 68 Risk, 276, 304, 337, 478, 485–487, 492, 494, 500, 501 assessment, 500 factor, 287 heterogeneity, 496 of infection, 478 vector, 479 Ross, Sir R.A., 6, 13, 229, 230, 391, 394, 509 Ross–MacDonald model, 393, 398 Routh–Hurwitz conditions, 33, 67 Row operation, 544 Rubella, 15, 22, 71, 429

S Saddle point, 88, 342, 468, 602–604 Sanitation, 100 SARS, 6, 14, 162, 216, 217, 337, 457, 461, 501, 509, 510 Saturation, 43 of contacts, 313 level, 277 Scalar, 540, 548 product, 540 Scarlet fever, 340 Schistomiasis, 21

Index Schistosomiasis, 3, 13, 229 Seasonal epidemic, 311, 315, 323, 326, 327, 331, 347 Seasonal forcing, 160 Seasonal influenza, 326, 347 Seasonality, 347 mosquito population, 394 Seasonal outbreak, 11, 323, 328, 332 Seasonal strain, 327, 329, 330, 348 Seasonal variation, 72, 334 Secondary cases, 120 Secondary infected vertex, 120 Secondary infection, 124, 164, 168, 182–185, 190, 220, 222, 277, 393, 463 Second wave, 330–334 Sensitive strain, 253, 269–271 Sensitivity, 189 analysis, 334 Separable, 563, 565, 569, 570 equation, 562 pair formation, 445, 446 Separation of variables, 27, 101 Separatrix, 88, 404, 604 Serial interval, 157 Seroconversion, 276, 304 Serological study, 39 Severity, 104, 312 Sexual activity, 273, 274, 286, 418 Sexual contact, 275, 303, 304, 418 Sexually active, 281, 286, 302, 303 population, 276, 304 population size, 282 Sexually transmitted diseases, 14, 23, 188, 429, 445 Sexual partner, 275–277, 304 Sexual transmission, 286 Sex workers, 274 Sickle cell carrier, 407 Sickle cell disease, 398, 406, 407 Sickle cell genetics, 398 Sickle cell trait, 405–407 Sigmoid function, 105 Sign structure, 74 Singular perturbation, 236–239, 243, 400 SIR model, 23, 53 SIS model, 23, 49, 52 Sleeping sickness, 3, 21 Slow dynamics, 402 Slower than exponential growth, 385 Slow manifold, 402 Slow progression, 261, 268 Slow system, 108 Slow variable, 108 Smallpox, 4, 5, 69, 477

Index Snow, John, 5 Social contact, 7, 118 Social interaction, 54 Sojourn distribution, 378, 379 Sojourn time, 163 Spatial distribution, 465 Spatial heterogeneity, 429 Spatial model, 458 Spatial pattern, 471 Spatial spread, 330, 464, 466, 509 Spatial structure, 384 Spectral bound, 185 Spectral radius, 184, 187, 205, 217, 324 Spiral, 596 point, 603, 604 Stability, 11, 51, 53, 77, 263, 277, 283, 340, 344, 345, 404, 584, 593, 595 of equilibrium, 133 index, 394 region, 77, 345 theorem, 593, 594 transfer, 275 Stable age distribution, 434, 435, 449, 451 Stable manifold, 404 Stage distribution, 361, 367, 371 Staged progress, 170 Staged progression, 273 Stage duration, 379 exponentially distributed, 268 Stages of infection, 316 Stage transition, 379 Standard incidence, 24, 43, 44, 283, 356, 404 Starting time, 333 Start time, 354 State at infection, 187 State equation, 258 Stay in compartment arbitrarily distributed, 7 exponentially distributed, 7 STD, 274, 445 Steady state, 449 Stochastic branching process model, 125 Stochastic event, 7, 118 Stochastic model, 7, 8, 15, 42, 122, 189, 354, 362 Stochastic phase, 10, 152, 154 Stochastic simulation, 312, 354 Strain, 251 Strategic model, 9, 330, 331, 333 Strategy, 321, 322 Strict barrier, 352 Subgroup, 275 Superinfection, 256, 347, 394 Superspreader, 14, 123, 188

617 Superspreading, 122 Sub-population, 189, 196, 199, 202, 204, 208, 210, 219, 221, 223, 266, 457 Surveillance, 354 Surveillance data, 331 Survival function, 365, 367, 369, 375, 377 Survival probability, 142, 372, 438 Survivorship function, 275, 279, 304 Susceptibility, 144, 179, 182, 190, 195, 312, 316, 319, 323, 327, 334, 347, 384 Susceptible, 5–7, 10, 22–24, 26, 28–30, 33, 35, 36, 39, 40, 44, 46, 48, 50, 52, 55–59, 64, 65, 67, 70, 73, 75, 78, 82, 83, 90, 100–102, 104, 107–109, 111, 117, 118, 179–181, 185, 189, 190, 195, 196, 202, 208, 210, 217, 221, 222, 230, 232, 235, 251, 261, 264, 266, 268, 270, 274–279, 281, 284, 287, 293–295, 297–302, 304, 312–314, 316–319, 323, 327, 328, 330, 333, 336, 338, 342, 344, 346, 352, 356, 358, 359, 362, 386, 393, 396, 411, 421, 439–441, 443, 447, 448, 458–462, 469, 473, 474, 477, 479–481, 483, 491, 492, 494, 509, 521–523, 582 Sustained oscillation, 34, 338, 341, 344 Symptom, 337 Symptomatic attack rate, 315 Symptomatic disease case, 318 System of differential equations, 539

T Tactical model, 9, 330, 331, 334 Taylor’s theorem, 603 TB-HIV coinfection, 287 Technology, 501 Temporal behavior, 357 Temporal evolution, 356 Threshold, 6, 22, 28, 33, 63, 120, 230, 232, 345, 418, 460, 485, 496 conditions, 398 Time evolution, 356 from onset to death, 375 rescaled, 402 scale, 243, 287, 400 since onset, 367, 368 Tipping point, 28, 509 Total infectivity, 324, 328 Toxoplasmosis, 107 Trace, 32, 33, 67, 74, 88, 133, 187, 188, 593–595

618 Trait, 404 Transcendental equation, 75 Transcritical bifurcation, 275 Transient subpopulations, 477 Transition, 126, 326, 352, 365, 367, 368, 371, 372 diagram, 162, 164, 221, 252, 361, 372, 376, 377 probability, 222 rate, 155, 219, 354, 371, 377 time, 365 Transmissibility, 123, 125, 273, 332 Transmission, 6, 140, 159, 167, 334, 360, 387, 399, 480, 485, 489, 491 coefficient, 324, 452, 460, 469, 471 diagram, 253 dynamics, 107, 263, 266, 274, 324, 359, 488, 496 fitness, 510 heterogeneity, 489 intervention, 125 mode, 273, 337 network, 490 probability, 190, 274, 278, 304, 420 rate, 166, 270, 276, 281, 304, 331, 334, 352, 354, 356, 360, 363, 397, 398, 487, 492 reduced, 352 seasonality, 347 seasonally forced, 159, 334 Transpose, 547 Transversality condition, 258 Travel, 386, 387, 460, 461, 509 rate, 459–461, 471 restriction, 464 Traveling wave, 464, 468–470 Treatment, 65, 131, 159, 195, 252, 253, 261, 262, 264, 266, 312, 321, 322, 407 class, 131, 132 effort, 260, 452 failure, 253, 254, 257, 262, 263 fraction, 318 model, 37, 144 period distribution, 144, 145 probability, 407 rate, 252, 267, 319, 322, 407 stage, 132 Triangular matrix, 544 Tuberculosis, 4, 13, 21, 22, 104, 230, 249–253, 256–258, 260, 261, 263, 265–269, 284, 337, 463, 480, 486, 501, 511 active, 252, 286, 290 antibiotic resistant, 252 with distributed delay, 268

Index drug-resistant, 262, 266, 270 with drug-resistant strains, 252 drug-sensitive, 270 dynamics, 291 with fast and slow progression, 260, 261 HIV coexistence, 288 HIV coinfection, 291 infective, 286 with optimal control, 256 prevalence, 290, 291 with reinfection, 263 resistant, 251–254, 256, 257, 259, 260, 266 resistant strain, 251 sensitive, 252, 253, 270 treatable, 284 treatment, 287 two-strain, 253, 266 Tuberculosis cases, secondary, 287 Tularemia, 13, 229 Two-strain model, 256, 269, 342 Typhoid, 5 Typhus, 3, 21

U Unbounded solution, 573, 584 Unstable, 11, 12, 26, 32–34, 63, 64, 77, 78, 80–82, 88, 91, 95, 100, 101, 109, 110, 130, 185, 186, 223, 254, 255, 265, 278, 280, 284, 288, 304, 339, 340, 342, 344, 345, 404, 406, 444, 445, 449, 450, 452, 460, 471, 583, 592, 594, 595, 597 Untreated host, 242

V Vaccination, 9, 11, 15, 22, 48, 49, 56, 57, 60, 65, 68, 69, 82–84, 113, 114, 131, 159–162, 176, 179, 180, 182, 186, 188, 192, 194, 196, 197, 199, 204, 205, 208, 210, 216, 223, 224, 243, 266, 299, 300, 312, 316, 318, 319, 321, 330–332, 335, 336, 347, 440–443, 451, 454, 510 pulse, 47, 48 Vaccine, 4, 21, 69, 83, 114, 134, 159, 161, 179, 202, 205, 208, 216, 224, 299, 300, 311, 316, 319, 330–332, 336, 351, 360, 414 Variable infectivity, 274 Variolation, 4 Vector, 12, 13, 229, 230, 241, 391, 421, 501, 539, 540, 592

Index control, 490 population, 9, 234, 393, 447, 492, 494, 500 population size, 394 transmission, 230, 417, 420 model, 391, 393, 414 reproduction number, 415 Vector-borne disease, 13, 229, 501 Vector-matrix notation, 598 Vector-transmitted disease, 230, 232, 391 Vertex, 7, 118, 119, 122, 459 Vertical transmission, 12, 67, 273, 274, 410, 414, 501 Viral agents, 22 Viral load, 359 Virus, 23, 332, 333 Vortex, 602 Vulnerability, 190, 210, 211 W Waiting time, 364, 365, 367, 371, 375, 379

619 exponential, 371 exponentially distributed, 366 Wave, 333, 471 West Nile fever, 13 West Nile virus, 13, 229, 230 Whooping cough, 71 Within-host system, 107, 108

Y Yellow fever, 13, 229, 230

Z Zika, 13, 414, 418, 419, 463, 480, 490, 501 transmission through sexual contact, 414 virus, 229 ZIKV, 490, 491, 493, 494, 496–501 Z-sign pattern, 185